Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.4s

Alternatives: 18

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t)))))
   (fabs (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh / ew) / Math.tan(t)));
	return Math.abs((((ew * Math.sin(t)) * Math.cos(t_1)) + ((eh * Math.cos(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh / ew) / math.tan(t)))
	return math.fabs((((ew * math.sin(t)) * math.cos(t_1)) + ((eh * math.cos(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t)))))
   (fabs (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh / ew) / Math.tan(t)));
	return Math.abs((((ew * Math.sin(t)) * Math.cos(t_1)) + ((eh * Math.cos(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh / ew) / math.tan(t)))
	return math.fabs((((ew * math.sin(t)) * math.cos(t_1)) + ((eh * math.cos(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \frac{t\_1}{\sin t \cdot ew}\\ \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {t\_2}^{2}}} \cdot ew, \sin t, t\_1 \cdot \tanh \sinh^{-1} t\_2\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)) (t_2 (/ t_1 (* (sin t) ew))))
   (fabs
    (fma
     (* (/ 1.0 (sqrt (+ 1.0 (pow t_2 2.0)))) ew)
     (sin t)
     (* t_1 (tanh (asinh t_2)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = t_1 / (sin(t) * ew);
	return fabs(fma(((1.0 / sqrt((1.0 + pow(t_2, 2.0)))) * ew), sin(t), (t_1 * tanh(asinh(t_2)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = Float64(t_1 / Float64(sin(t) * ew))
	return abs(fma(Float64(Float64(1.0 / sqrt(Float64(1.0 + (t_2 ^ 2.0)))) * ew), sin(t), Float64(t_1 * tanh(asinh(t_2)))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(1.0 / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(t$95$1 * N[Tanh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

2. Taylor expanded in eh around 0

   \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

   2. associate-*r* N/A

      \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

   3. lower-fma.f64 N/A

      \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t)))))
   (fabs (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh / ew) / Math.tan(t)));
	return Math.abs((((ew * Math.sin(t)) * Math.cos(t_1)) + ((eh * Math.cos(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh / ew) / math.tan(t)))
	return math.fabs((((ew * math.sin(t)) * math.cos(t_1)) + ((eh * math.cos(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. Add Preprocessing

Alternative 3: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot \tan t}\\ \left|\mathsf{fma}\left(\sin t \cdot ew, \frac{1}{\sqrt{1 + {t\_1}^{2}}}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} t\_1\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (fabs
    (fma
     (* (sin t) ew)
     (/ 1.0 (sqrt (+ 1.0 (pow t_1 2.0))))
     (* (* (cos t) eh) (tanh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	return fabs(fma((sin(t) * ew), (1.0 / sqrt((1.0 + pow(t_1, 2.0)))), ((cos(t) * eh) * tanh(asinh(t_1)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	return abs(fma(Float64(sin(t) * ew), Float64(1.0 / sqrt(Float64(1.0 + (t_1 ^ 2.0)))), Float64(Float64(cos(t) * eh) * tanh(asinh(t_1)))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\sin t \cdot ew, \frac{1}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}\right| \]
4. Add Preprocessing

Alternative 4: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ \left|\mathsf{fma}\left(ew, \sin t, t\_1 \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{\sin t \cdot ew}\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)))
   (fabs (fma ew (sin t) (* t_1 (tanh (asinh (/ t_1 (* (sin t) ew)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	return fabs(fma(ew, sin(t), (t_1 * tanh(asinh((t_1 / (sin(t) * ew)))))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	return abs(fma(ew, sin(t), Float64(t_1 * tanh(asinh(Float64(t_1 / Float64(sin(t) * ew)))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(t$95$1 * N[Tanh[N[ArcSinh[N[(t$95$1 / N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

2. Taylor expanded in eh around 0

   \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

   2. associate-*r* N/A

      \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

   3. lower-fma.f64 N/A

      \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

5. Taylor expanded in eh around 0

   \[\leadsto \left|\mathsf{fma}\left(ew, \sin \color{blue}{t}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
6. Step-by-step derivation

7. Applied rewrites 98.5%

   \[\leadsto \left|\mathsf{fma}\left(ew, \sin \color{blue}{t}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
8. Add Preprocessing

Alternative 5: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := ew \cdot \sin t\\ t_3 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_4 := \frac{eh}{t \cdot ew}\\ \mathbf{if}\;\left|t\_2 \cdot \cos t\_3 + t\_1 \cdot \sin t\_3\right| \leq 5 \cdot 10^{+42}:\\ \;\;\;\;\left|\mathsf{fma}\left(1 \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_2 \cdot \frac{1}{\sqrt{1 + t\_4 \cdot t\_4}} + t\_1 \cdot \sin \tan^{-1} t\_4\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t)))
        (t_2 (* ew (sin t)))
        (t_3 (atan (/ (/ eh ew) (tan t))))
        (t_4 (/ eh (* t ew))))
   (if (<= (fabs (+ (* t_2 (cos t_3)) (* t_1 (sin t_3)))) 5e+42)
     (fabs
      (fma
       (* 1.0 ew)
       (sin t)
       (*
        (* (cos t) eh)
        (tanh
         (asinh
          (/
           (fma
            (* t t)
            (- (* -0.5 (/ eh ew)) (* -0.16666666666666666 (/ eh ew)))
            (/ eh ew))
           t))))))
     (fabs
      (+
       (* t_2 (/ 1.0 (sqrt (+ 1.0 (* t_4 t_4)))))
       (* t_1 (sin (atan t_4))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = ew * sin(t);
	double t_3 = atan(((eh / ew) / tan(t)));
	double t_4 = eh / (t * ew);
	double tmp;
	if (fabs(((t_2 * cos(t_3)) + (t_1 * sin(t_3)))) <= 5e+42) {
		tmp = fabs(fma((1.0 * ew), sin(t), ((cos(t) * eh) * tanh(asinh((fma((t * t), ((-0.5 * (eh / ew)) - (-0.16666666666666666 * (eh / ew))), (eh / ew)) / t))))));
	} else {
		tmp = fabs(((t_2 * (1.0 / sqrt((1.0 + (t_4 * t_4))))) + (t_1 * sin(atan(t_4)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = Float64(ew * sin(t))
	t_3 = atan(Float64(Float64(eh / ew) / tan(t)))
	t_4 = Float64(eh / Float64(t * ew))
	tmp = 0.0
	if (abs(Float64(Float64(t_2 * cos(t_3)) + Float64(t_1 * sin(t_3)))) <= 5e+42)
		tmp = abs(fma(Float64(1.0 * ew), sin(t), Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(fma(Float64(t * t), Float64(Float64(-0.5 * Float64(eh / ew)) - Float64(-0.16666666666666666 * Float64(eh / ew))), Float64(eh / ew)) / t))))));
	else
		tmp = abs(Float64(Float64(t_2 * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_4 * t_4))))) + Float64(t_1 * sin(atan(t_4)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e+42], N[Abs[N[(N[(1.0 * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$2 * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Sin[N[ArcTan[t$95$4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))) < 5.00000000000000007e42

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      3. unpow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      5. lower--.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      7. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      9. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      10. lift-/.f64 72.3

          \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   7. Applied rewrites 72.3%

      \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)\right)\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      3. unpow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      5. lower--.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      7. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      9. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      10. lift-/.f64 72.4

          \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

   10. Applied rewrites 72.4%

       \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

   11. Taylor expanded in eh around 0

       \[\leadsto \left|\mathsf{fma}\left(1 \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 82.9%

       \[\leadsto \left|\mathsf{fma}\left(1 \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

   if 5.00000000000000007e42 < (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))))

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in t around 0

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{\color{blue}{ew \cdot t}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot \color{blue}{ew}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      3. lower-*.f64 99.1

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot \color{blue}{ew}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   4. Applied rewrites 99.1%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right| \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\color{blue}{ew \cdot t}}\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot \color{blue}{ew}}\right)\right| \]

      3. lower-*.f64 89.9

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot \color{blue}{ew}}\right)\right| \]

   7. Applied rewrites 89.9%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{t \cdot ew}\right)}\right| \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \color{blue}{\cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \color{blue}{\tan^{-1} \left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right| \]

      3. cos-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\color{blue}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right| \]

      6. lower-+.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{\color{blue}{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right| \]

      7. lower-*.f64 89.9

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + \color{blue}{\frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right| \]

   9. Applied rewrites 89.9%

      \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + \frac{eh}{t \cdot ew} \cdot \frac{eh}{t \cdot ew}}}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ \mathbf{if}\;eh \leq 1.5 \cdot 10^{+189}:\\ \;\;\;\;\left|\mathsf{fma}\left(1 \cdot ew, \sin t, t\_1 \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, t, t\_1 \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)))
   (if (<= eh 1.5e+189)
     (fabs
      (fma
       (* 1.0 ew)
       (sin t)
       (*
        t_1
        (tanh
         (asinh
          (/
           (fma
            (* t t)
            (- (* -0.5 (/ eh ew)) (* -0.16666666666666666 (/ eh ew)))
            (/ eh ew))
           t))))))
     (fabs
      (fma
       ew
       t
       (*
        t_1
        (tanh
         (+
          (log (+ (/ (fabs eh) (sqrt (* ew ew))) (/ eh ew)))
          (* -1.0 (log t))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double tmp;
	if (eh <= 1.5e+189) {
		tmp = fabs(fma((1.0 * ew), sin(t), (t_1 * tanh(asinh((fma((t * t), ((-0.5 * (eh / ew)) - (-0.16666666666666666 * (eh / ew))), (eh / ew)) / t))))));
	} else {
		tmp = fabs(fma(ew, t, (t_1 * tanh((log(((fabs(eh) / sqrt((ew * ew))) + (eh / ew))) + (-1.0 * log(t)))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	tmp = 0.0
	if (eh <= 1.5e+189)
		tmp = abs(fma(Float64(1.0 * ew), sin(t), Float64(t_1 * tanh(asinh(Float64(fma(Float64(t * t), Float64(Float64(-0.5 * Float64(eh / ew)) - Float64(-0.16666666666666666 * Float64(eh / ew))), Float64(eh / ew)) / t))))));
	else
		tmp = abs(fma(ew, t, Float64(t_1 * tanh(Float64(log(Float64(Float64(abs(eh) / sqrt(Float64(ew * ew))) + Float64(eh / ew))) + Float64(-1.0 * log(t)))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, 1.5e+189], N[Abs[N[(N[(1.0 * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(t$95$1 * N[Tanh[N[ArcSinh[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * t + N[(t$95$1 * N[Tanh[N[(N[Log[N[(N[(N[Abs[eh], $MachinePrecision] / N[Sqrt[N[(ew * ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.4999999999999999e189

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      3. unpow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      5. lower--.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      7. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      9. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

      10. lift-/.f64 72.3

          \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   7. Applied rewrites 72.3%

      \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)\right)\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      3. unpow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      5. lower--.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      7. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      9. lift-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

      10. lift-/.f64 72.4

          \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

   10. Applied rewrites 72.4%

       \[\leadsto \left|\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

   11. Taylor expanded in eh around 0

       \[\leadsto \left|\mathsf{fma}\left(1 \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 82.9%

       \[\leadsto \left|\mathsf{fma}\left(1 \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right| \]

   if 1.4999999999999999e189 < eh

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|\mathsf{fma}\left(ew, \sin \color{blue}{t}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sin \color{blue}{t}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 64.9%

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   11. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{t \cdot ew}\right)\right)\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 64.9%

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{t \cdot ew}\right)\right)\right| \]

   14. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       2. lower-log.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       3. lower-+.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       4. sqrt-div N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\sqrt{{eh}^{2}}}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       5. lower-/.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\sqrt{{eh}^{2}}}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       6. pow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\sqrt{eh \cdot eh}}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       7. rem-sqrt-square N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       8. lower-fabs.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       10. pow2 N/A

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       11. lower-*.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       12. lift-/.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       13. lower-*.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       14. lower-log.f64 26.6

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

   16. Applied rewrites 26.6%

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ \mathbf{if}\;t \leq 0.0215:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, t, t\_1 \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{t \cdot ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)))
   (if (<= t 0.0215)
     (fabs (fma ew t (* t_1 (tanh (asinh (/ t_1 (* t ew)))))))
     (fabs (* ew (sin t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double tmp;
	if (t <= 0.0215) {
		tmp = fabs(fma(ew, t, (t_1 * tanh(asinh((t_1 / (t * ew)))))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	tmp = 0.0
	if (t <= 0.0215)
		tmp = abs(fma(ew, t, Float64(t_1 * tanh(asinh(Float64(t_1 / Float64(t * ew)))))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[t, 0.0215], N[Abs[N[(ew * t + N[(t$95$1 * N[Tanh[N[ArcSinh[N[(t$95$1 / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 0.021499999999999998

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|\mathsf{fma}\left(ew, \sin \color{blue}{t}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sin \color{blue}{t}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 64.9%

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   11. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{t \cdot ew}\right)\right)\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 64.9%

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{t \cdot ew}\right)\right)\right| \]

   if 0.021499999999999998 < t

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 50.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0215:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t 0.0215)
   (fabs
    (fma
     ew
     t
     (*
      (* (cos t) eh)
      (tanh
       (+
        (log (+ (/ (fabs eh) (sqrt (* ew ew))) (/ eh ew)))
        (* -1.0 (log t)))))))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 0.0215) {
		tmp = fabs(fma(ew, t, ((cos(t) * eh) * tanh((log(((fabs(eh) / sqrt((ew * ew))) + (eh / ew))) + (-1.0 * log(t)))))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= 0.0215)
		tmp = abs(fma(ew, t, Float64(Float64(cos(t) * eh) * tanh(Float64(log(Float64(Float64(abs(eh) / sqrt(Float64(ew * ew))) + Float64(eh / ew))) + Float64(-1.0 * log(t)))))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, 0.0215], N[Abs[N[(ew * t + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[(N[Log[N[(N[(N[Abs[eh], $MachinePrecision] / N[Sqrt[N[(ew * ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.021499999999999998

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|\mathsf{fma}\left(ew, \sin \color{blue}{t}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sin \color{blue}{t}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 64.9%

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)\right| \]

   11. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{t \cdot ew}\right)\right)\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 64.9%

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{t \cdot ew}\right)\right)\right| \]

   14. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       2. lower-log.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       3. lower-+.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       4. sqrt-div N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\sqrt{{eh}^{2}}}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       5. lower-/.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\sqrt{{eh}^{2}}}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       6. pow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\sqrt{eh \cdot eh}}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       7. rem-sqrt-square N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       8. lower-fabs.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{{ew}^{2}}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       10. pow2 N/A

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       11. lower-*.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       12. lift-/.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       13. lower-*.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

       14. lower-log.f64 26.6

           \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

   16. Applied rewrites 26.6%

       \[\leadsto \left|\mathsf{fma}\left(ew, t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(\frac{\left|eh\right|}{\sqrt{ew \cdot ew}} + \frac{eh}{ew}\right) + -1 \cdot \log t\right)\right)\right| \]

   if 0.021499999999999998 < t

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 38.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-62}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\left(1 + -0.5 \cdot \left(t \cdot t\right)\right) \cdot eh}{t \cdot ew}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t 6.8e-62)
   (fabs (* (tanh (asinh (/ (* (+ 1.0 (* -0.5 (* t t))) eh) (* t ew)))) eh))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 6.8e-62) {
		tmp = fabs((tanh(asinh((((1.0 + (-0.5 * (t * t))) * eh) / (t * ew)))) * eh));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= 6.8e-62:
		tmp = math.fabs((math.tanh(math.asinh((((1.0 + (-0.5 * (t * t))) * eh) / (t * ew)))) * eh))
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= 6.8e-62)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(Float64(1.0 + Float64(-0.5 * Float64(t * t))) * eh) / Float64(t * ew)))) * eh));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= 6.8e-62)
		tmp = abs((tanh(asinh((((1.0 + (-0.5 * (t * t))) * eh) / (t * ew)))) * eh));
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, 6.8e-62], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(N[(1.0 + N[(-0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.79999999999999975e-62

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)}\right| \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \color{blue}{eh}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \color{blue}{eh}\right| \]

   4. Applied rewrites 40.9%

      \[\leadsto \left|\color{blue}{\tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right) \cdot eh}\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\left(1 + \frac{-1}{2} \cdot {t}^{2}\right) \cdot eh}{\sin t \cdot ew}\right) \cdot eh\right| \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\left(1 + \frac{-1}{2} \cdot {t}^{2}\right) \cdot eh}{\sin t \cdot ew}\right) \cdot eh\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\left(1 + \frac{-1}{2} \cdot {t}^{2}\right) \cdot eh}{\sin t \cdot ew}\right) \cdot eh\right| \]

      3. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\left(1 + \frac{-1}{2} \cdot \left(t \cdot t\right)\right) \cdot eh}{\sin t \cdot ew}\right) \cdot eh\right| \]

      4. lower-*.f64 41.1

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\left(1 + -0.5 \cdot \left(t \cdot t\right)\right) \cdot eh}{\sin t \cdot ew}\right) \cdot eh\right| \]

   7. Applied rewrites 41.1%

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\left(1 + -0.5 \cdot \left(t \cdot t\right)\right) \cdot eh}{\sin t \cdot ew}\right) \cdot eh\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\left(1 + \frac{-1}{2} \cdot \left(t \cdot t\right)\right) \cdot eh}{t \cdot ew}\right) \cdot eh\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 40.3%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\left(1 + -0.5 \cdot \left(t \cdot t\right)\right) \cdot eh}{t \cdot ew}\right) \cdot eh\right| \]

   if 6.79999999999999975e-62 < t

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 33.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\left|\frac{eh \cdot eh}{\left|eh\right|}\right|\\ \mathbf{elif}\;ew \leq 1.86 \cdot 10^{-38}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 7.5e-155)
   (fabs (/ (* eh eh) (fabs eh)))
   (if (<= ew 1.86e-38)
     (fabs (/ eh (* ew (sqrt (/ 1.0 (* ew ew))))))
     (fabs (* ew (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = fabs(((eh * eh) / fabs(eh)));
	} else if (ew <= 1.86e-38) {
		tmp = fabs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 7.5d-155) then
        tmp = abs(((eh * eh) / abs(eh)))
    else if (ew <= 1.86d-38) then
        tmp = abs((eh / (ew * sqrt((1.0d0 / (ew * ew))))))
    else
        tmp = abs((ew * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = Math.abs(((eh * eh) / Math.abs(eh)));
	} else if (ew <= 1.86e-38) {
		tmp = Math.abs((eh / (ew * Math.sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = Math.abs((ew * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 7.5e-155:
		tmp = math.fabs(((eh * eh) / math.fabs(eh)))
	elif ew <= 1.86e-38:
		tmp = math.fabs((eh / (ew * math.sqrt((1.0 / (ew * ew))))))
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 7.5e-155)
		tmp = abs(Float64(Float64(eh * eh) / abs(eh)));
	elseif (ew <= 1.86e-38)
		tmp = abs(Float64(eh / Float64(ew * sqrt(Float64(1.0 / Float64(ew * ew))))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 7.5e-155)
		tmp = abs(((eh * eh) / abs(eh)));
	elseif (ew <= 1.86e-38)
		tmp = abs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 7.5e-155], N[Abs[N[(N[(eh * eh), $MachinePrecision] / N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.86e-38], N[Abs[N[(eh / N[(ew * N[Sqrt[N[(1.0 / N[(ew * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < 7.5000000000000006e-155

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

      3. lift-*.f64 21.5

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

   9. Applied rewrites 21.5%

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       3. rem-sqrt-square N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

       4. lower-fabs.f64 24.2

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   11. Applied rewrites 24.2%

       \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   if 7.5000000000000006e-155 < ew < 1.85999999999999995e-38

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in eh around 0

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

      6. lift-*.f64 24.1

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

   9. Applied rewrites 24.1%

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}}\right| \]

   if 1.85999999999999995e-38 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\left|\frac{eh \cdot eh}{\left|eh\right|}\right|\\ \mathbf{elif}\;ew \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(t \cdot t\right)\right) - 0.16666666666666666\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 7.5e-155)
   (fabs (/ (* eh eh) (fabs eh)))
   (if (<= ew 4e+110)
     (fabs (/ eh (* ew (sqrt (/ 1.0 (* ew ew))))))
     (fabs
      (*
       ew
       (*
        t
        (+
         1.0
         (*
          (* t t)
          (-
           (*
            (* t t)
            (+ 0.008333333333333333 (* -0.0001984126984126984 (* t t))))
           0.16666666666666666)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = fabs(((eh * eh) / fabs(eh)));
	} else if (ew <= 4e+110) {
		tmp = fabs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = fabs((ew * (t * (1.0 + ((t * t) * (((t * t) * (0.008333333333333333 + (-0.0001984126984126984 * (t * t)))) - 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 7.5d-155) then
        tmp = abs(((eh * eh) / abs(eh)))
    else if (ew <= 4d+110) then
        tmp = abs((eh / (ew * sqrt((1.0d0 / (ew * ew))))))
    else
        tmp = abs((ew * (t * (1.0d0 + ((t * t) * (((t * t) * (0.008333333333333333d0 + ((-0.0001984126984126984d0) * (t * t)))) - 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = Math.abs(((eh * eh) / Math.abs(eh)));
	} else if (ew <= 4e+110) {
		tmp = Math.abs((eh / (ew * Math.sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = Math.abs((ew * (t * (1.0 + ((t * t) * (((t * t) * (0.008333333333333333 + (-0.0001984126984126984 * (t * t)))) - 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 7.5e-155:
		tmp = math.fabs(((eh * eh) / math.fabs(eh)))
	elif ew <= 4e+110:
		tmp = math.fabs((eh / (ew * math.sqrt((1.0 / (ew * ew))))))
	else:
		tmp = math.fabs((ew * (t * (1.0 + ((t * t) * (((t * t) * (0.008333333333333333 + (-0.0001984126984126984 * (t * t)))) - 0.16666666666666666))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 7.5e-155)
		tmp = abs(Float64(Float64(eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs(Float64(eh / Float64(ew * sqrt(Float64(1.0 / Float64(ew * ew))))));
	else
		tmp = abs(Float64(ew * Float64(t * Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(Float64(t * t) * Float64(0.008333333333333333 + Float64(-0.0001984126984126984 * Float64(t * t)))) - 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 7.5e-155)
		tmp = abs(((eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	else
		tmp = abs((ew * (t * (1.0 + ((t * t) * (((t * t) * (0.008333333333333333 + (-0.0001984126984126984 * (t * t)))) - 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 7.5e-155], N[Abs[N[(N[(eh * eh), $MachinePrecision] / N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4e+110], N[Abs[N[(eh / N[(ew * N[Sqrt[N[(1.0 / N[(ew * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(t * N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * N[(0.008333333333333333 + N[(-0.0001984126984126984 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < 7.5000000000000006e-155

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

      3. lift-*.f64 21.5

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

   9. Applied rewrites 21.5%

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       3. rem-sqrt-square N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

       4. lower-fabs.f64 24.2

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   11. Applied rewrites 24.2%

       \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   if 7.5000000000000006e-155 < ew < 4.0000000000000001e110

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in eh around 0

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

      6. lift-*.f64 24.1

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

   9. Applied rewrites 24.1%

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}}\right| \]

   if 4.0000000000000001e110 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \color{blue}{{t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right| \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + {t}^{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + {t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + {t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      4. pow2 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      5. lift-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      6. lower--.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      7. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      8. pow2 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      9. lift-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      10. lower-+.f64 N/A

          \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      11. lower-*.f64 N/A

          \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)\right)\right| \]

      12. pow2 N/A

          \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot \left(t \cdot t\right)\right) - \frac{1}{6}\right)\right)\right)\right| \]

      13. lift-*.f64 18.6

          \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(t \cdot t\right)\right) - 0.16666666666666666\right)\right)\right)\right| \]

   10. Applied rewrites 18.6%

       \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \color{blue}{\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(t \cdot t\right)\right) - 0.16666666666666666\right)}\right)\right)\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 31.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\left|\frac{eh \cdot eh}{\left|eh\right|}\right|\\ \mathbf{elif}\;ew \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot \left(ew + \left(t \cdot t\right) \cdot \mathsf{fma}\left(-0.16666666666666666, ew, 0.008333333333333333 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 7.5e-155)
   (fabs (/ (* eh eh) (fabs eh)))
   (if (<= ew 4e+110)
     (fabs (/ eh (* ew (sqrt (/ 1.0 (* ew ew))))))
     (fabs
      (*
       t
       (+
        ew
        (*
         (* t t)
         (fma
          -0.16666666666666666
          ew
          (* 0.008333333333333333 (* ew (* t t)))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = fabs(((eh * eh) / fabs(eh)));
	} else if (ew <= 4e+110) {
		tmp = fabs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = fabs((t * (ew + ((t * t) * fma(-0.16666666666666666, ew, (0.008333333333333333 * (ew * (t * t))))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 7.5e-155)
		tmp = abs(Float64(Float64(eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs(Float64(eh / Float64(ew * sqrt(Float64(1.0 / Float64(ew * ew))))));
	else
		tmp = abs(Float64(t * Float64(ew + Float64(Float64(t * t) * fma(-0.16666666666666666, ew, Float64(0.008333333333333333 * Float64(ew * Float64(t * t))))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 7.5e-155], N[Abs[N[(N[(eh * eh), $MachinePrecision] / N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4e+110], N[Abs[N[(eh / N[(ew * N[Sqrt[N[(1.0 / N[(ew * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t * N[(ew + N[(N[(t * t), $MachinePrecision] * N[(-0.16666666666666666 * ew + N[(0.008333333333333333 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < 7.5000000000000006e-155

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

      3. lift-*.f64 21.5

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

   9. Applied rewrites 21.5%

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       3. rem-sqrt-square N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

       4. lower-fabs.f64 24.2

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   11. Applied rewrites 24.2%

       \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   if 7.5000000000000006e-155 < ew < 4.0000000000000001e110

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in eh around 0

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

      6. lift-*.f64 24.1

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

   9. Applied rewrites 24.1%

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}}\right| \]

   if 4.0000000000000001e110 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|t \cdot \left(ew + \color{blue}{{t}^{2} \cdot \left(\frac{-1}{6} \cdot ew + \frac{1}{120} \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)\right| \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + {t}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot ew + \frac{1}{120} \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + {t}^{2} \cdot \left(\frac{-1}{6} \cdot ew + \color{blue}{\frac{1}{120} \cdot \left(ew \cdot {t}^{2}\right)}\right)\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + {t}^{2} \cdot \left(\frac{-1}{6} \cdot ew + \frac{1}{120} \cdot \color{blue}{\left(ew \cdot {t}^{2}\right)}\right)\right)\right| \]

      4. pow2 N/A

         \[\leadsto \left|t \cdot \left(ew + \left(t \cdot t\right) \cdot \left(\frac{-1}{6} \cdot ew + \frac{1}{120} \cdot \left(\color{blue}{ew} \cdot {t}^{2}\right)\right)\right)\right| \]

      5. lift-*.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + \left(t \cdot t\right) \cdot \left(\frac{-1}{6} \cdot ew + \frac{1}{120} \cdot \left(\color{blue}{ew} \cdot {t}^{2}\right)\right)\right)\right| \]

      6. lower-fma.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + \left(t \cdot t\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, ew, \frac{1}{120} \cdot \left(ew \cdot {t}^{2}\right)\right)\right)\right| \]

      7. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + \left(t \cdot t\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, ew, \frac{1}{120} \cdot \left(ew \cdot {t}^{2}\right)\right)\right)\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + \left(t \cdot t\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, ew, \frac{1}{120} \cdot \left(ew \cdot {t}^{2}\right)\right)\right)\right| \]

      9. pow2 N/A

         \[\leadsto \left|t \cdot \left(ew + \left(t \cdot t\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, ew, \frac{1}{120} \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right)\right| \]

      10. lift-*.f64 18.6

          \[\leadsto \left|t \cdot \left(ew + \left(t \cdot t\right) \cdot \mathsf{fma}\left(-0.16666666666666666, ew, 0.008333333333333333 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right)\right| \]

   10. Applied rewrites 18.6%

       \[\leadsto \left|t \cdot \left(ew + \color{blue}{\left(t \cdot t\right) \cdot \mathsf{fma}\left(-0.16666666666666666, ew, 0.008333333333333333 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)}\right)\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 31.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\left|\frac{eh \cdot eh}{\left|eh\right|}\right|\\ \mathbf{elif}\;ew \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 7.5e-155)
   (fabs (/ (* eh eh) (fabs eh)))
   (if (<= ew 4e+110)
     (fabs (/ eh (* ew (sqrt (/ 1.0 (* ew ew))))))
     (fabs
      (*
       ew
       (*
        t
        (+
         1.0
         (*
          (* t t)
          (- (* 0.008333333333333333 (* t t)) 0.16666666666666666)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = fabs(((eh * eh) / fabs(eh)));
	} else if (ew <= 4e+110) {
		tmp = fabs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = fabs((ew * (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 7.5d-155) then
        tmp = abs(((eh * eh) / abs(eh)))
    else if (ew <= 4d+110) then
        tmp = abs((eh / (ew * sqrt((1.0d0 / (ew * ew))))))
    else
        tmp = abs((ew * (t * (1.0d0 + ((t * t) * ((0.008333333333333333d0 * (t * t)) - 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = Math.abs(((eh * eh) / Math.abs(eh)));
	} else if (ew <= 4e+110) {
		tmp = Math.abs((eh / (ew * Math.sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = Math.abs((ew * (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 7.5e-155:
		tmp = math.fabs(((eh * eh) / math.fabs(eh)))
	elif ew <= 4e+110:
		tmp = math.fabs((eh / (ew * math.sqrt((1.0 / (ew * ew))))))
	else:
		tmp = math.fabs((ew * (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 7.5e-155)
		tmp = abs(Float64(Float64(eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs(Float64(eh / Float64(ew * sqrt(Float64(1.0 / Float64(ew * ew))))));
	else
		tmp = abs(Float64(ew * Float64(t * Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(0.008333333333333333 * Float64(t * t)) - 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 7.5e-155)
		tmp = abs(((eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	else
		tmp = abs((ew * (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 7.5e-155], N[Abs[N[(N[(eh * eh), $MachinePrecision] / N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4e+110], N[Abs[N[(eh / N[(ew * N[Sqrt[N[(1.0 / N[(ew * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(t * N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(t * t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < 7.5000000000000006e-155

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

      3. lift-*.f64 21.5

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

   9. Applied rewrites 21.5%

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       3. rem-sqrt-square N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

       4. lower-fabs.f64 24.2

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   11. Applied rewrites 24.2%

       \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   if 7.5000000000000006e-155 < ew < 4.0000000000000001e110

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in eh around 0

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

      6. lift-*.f64 24.1

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

   9. Applied rewrites 24.1%

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}}\right| \]

   if 4.0000000000000001e110 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \color{blue}{{t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)}\right)\right)\right| \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + {t}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)}\right)\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \color{blue}{\frac{1}{6}}\right)\right)\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)\right| \]

      4. pow2 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)\right| \]

      5. lift-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)\right| \]

      6. lower--.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)\right| \]

      7. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)\right| \]

      8. pow2 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)\right)\right| \]

      9. lift-*.f64 18.6

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)\right)\right| \]

   10. Applied rewrites 18.6%

       \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \color{blue}{\left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)}\right)\right)\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 31.3% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\left|\frac{eh \cdot eh}{\left|eh\right|}\right|\\ \mathbf{elif}\;ew \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(t \cdot \left(1 + -0.16666666666666666 \cdot \left(t \cdot t\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 7.5e-155)
   (fabs (/ (* eh eh) (fabs eh)))
   (if (<= ew 4e+110)
     (fabs (/ eh (* ew (sqrt (/ 1.0 (* ew ew))))))
     (fabs (* ew (* t (+ 1.0 (* -0.16666666666666666 (* t t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = fabs(((eh * eh) / fabs(eh)));
	} else if (ew <= 4e+110) {
		tmp = fabs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = fabs((ew * (t * (1.0 + (-0.16666666666666666 * (t * t))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 7.5d-155) then
        tmp = abs(((eh * eh) / abs(eh)))
    else if (ew <= 4d+110) then
        tmp = abs((eh / (ew * sqrt((1.0d0 / (ew * ew))))))
    else
        tmp = abs((ew * (t * (1.0d0 + ((-0.16666666666666666d0) * (t * t))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = Math.abs(((eh * eh) / Math.abs(eh)));
	} else if (ew <= 4e+110) {
		tmp = Math.abs((eh / (ew * Math.sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = Math.abs((ew * (t * (1.0 + (-0.16666666666666666 * (t * t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 7.5e-155:
		tmp = math.fabs(((eh * eh) / math.fabs(eh)))
	elif ew <= 4e+110:
		tmp = math.fabs((eh / (ew * math.sqrt((1.0 / (ew * ew))))))
	else:
		tmp = math.fabs((ew * (t * (1.0 + (-0.16666666666666666 * (t * t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 7.5e-155)
		tmp = abs(Float64(Float64(eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs(Float64(eh / Float64(ew * sqrt(Float64(1.0 / Float64(ew * ew))))));
	else
		tmp = abs(Float64(ew * Float64(t * Float64(1.0 + Float64(-0.16666666666666666 * Float64(t * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 7.5e-155)
		tmp = abs(((eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	else
		tmp = abs((ew * (t * (1.0 + (-0.16666666666666666 * (t * t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 7.5e-155], N[Abs[N[(N[(eh * eh), $MachinePrecision] / N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4e+110], N[Abs[N[(eh / N[(ew * N[Sqrt[N[(1.0 / N[(ew * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(t * N[(1.0 + N[(-0.16666666666666666 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < 7.5000000000000006e-155

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

      3. lift-*.f64 21.5

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

   9. Applied rewrites 21.5%

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       3. rem-sqrt-square N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

       4. lower-fabs.f64 24.2

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   11. Applied rewrites 24.2%

       \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   if 7.5000000000000006e-155 < ew < 4.0000000000000001e110

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in eh around 0

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

      6. lift-*.f64 24.1

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

   9. Applied rewrites 24.1%

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}}\right| \]

   if 4.0000000000000001e110 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {t}^{2}}\right)\right)\right| \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{t}^{2}}\right)\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \frac{-1}{6} \cdot {t}^{\color{blue}{2}}\right)\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \frac{-1}{6} \cdot {t}^{2}\right)\right)\right| \]

      4. pow2 N/A

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \frac{-1}{6} \cdot \left(t \cdot t\right)\right)\right)\right| \]

      5. lift-*.f64 18.6

         \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + -0.16666666666666666 \cdot \left(t \cdot t\right)\right)\right)\right| \]

   10. Applied rewrites 18.6%

       \[\leadsto \left|ew \cdot \left(t \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot \left(t \cdot t\right)}\right)\right)\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 31.3% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\left|\frac{eh \cdot eh}{\left|eh\right|}\right|\\ \mathbf{elif}\;ew \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 7.5e-155)
   (fabs (/ (* eh eh) (fabs eh)))
   (if (<= ew 4e+110)
     (fabs (/ eh (* ew (sqrt (/ 1.0 (* ew ew))))))
     (fabs (* t (+ ew (* -0.16666666666666666 (* ew (* t t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = fabs(((eh * eh) / fabs(eh)));
	} else if (ew <= 4e+110) {
		tmp = fabs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = fabs((t * (ew + (-0.16666666666666666 * (ew * (t * t))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 7.5d-155) then
        tmp = abs(((eh * eh) / abs(eh)))
    else if (ew <= 4d+110) then
        tmp = abs((eh / (ew * sqrt((1.0d0 / (ew * ew))))))
    else
        tmp = abs((t * (ew + ((-0.16666666666666666d0) * (ew * (t * t))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = Math.abs(((eh * eh) / Math.abs(eh)));
	} else if (ew <= 4e+110) {
		tmp = Math.abs((eh / (ew * Math.sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = Math.abs((t * (ew + (-0.16666666666666666 * (ew * (t * t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 7.5e-155:
		tmp = math.fabs(((eh * eh) / math.fabs(eh)))
	elif ew <= 4e+110:
		tmp = math.fabs((eh / (ew * math.sqrt((1.0 / (ew * ew))))))
	else:
		tmp = math.fabs((t * (ew + (-0.16666666666666666 * (ew * (t * t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 7.5e-155)
		tmp = abs(Float64(Float64(eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs(Float64(eh / Float64(ew * sqrt(Float64(1.0 / Float64(ew * ew))))));
	else
		tmp = abs(Float64(t * Float64(ew + Float64(-0.16666666666666666 * Float64(ew * Float64(t * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 7.5e-155)
		tmp = abs(((eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	else
		tmp = abs((t * (ew + (-0.16666666666666666 * (ew * (t * t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 7.5e-155], N[Abs[N[(N[(eh * eh), $MachinePrecision] / N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4e+110], N[Abs[N[(eh / N[(ew * N[Sqrt[N[(1.0 / N[(ew * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t * N[(ew + N[(-0.16666666666666666 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < 7.5000000000000006e-155

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

      3. lift-*.f64 21.5

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

   9. Applied rewrites 21.5%

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       3. rem-sqrt-square N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

       4. lower-fabs.f64 24.2

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   11. Applied rewrites 24.2%

       \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   if 7.5000000000000006e-155 < ew < 4.0000000000000001e110

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in eh around 0

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

      6. lift-*.f64 24.1

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

   9. Applied rewrites 24.1%

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}}\right| \]

   if 4.0000000000000001e110 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|t \cdot \left(ew + \color{blue}{\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right)}\right)\right| \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + \frac{-1}{6} \cdot \color{blue}{\left(ew \cdot {t}^{2}\right)}\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + \frac{-1}{6} \cdot \left(ew \cdot \color{blue}{{t}^{2}}\right)\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + \frac{-1}{6} \cdot \left(ew \cdot {t}^{\color{blue}{2}}\right)\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(ew + \frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right)\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|t \cdot \left(ew + \frac{-1}{6} \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right| \]

      6. lift-*.f64 18.7

         \[\leadsto \left|t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right| \]

   10. Applied rewrites 18.7%

       \[\leadsto \left|t \cdot \left(ew + \color{blue}{-0.16666666666666666 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 31.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\left|\frac{eh \cdot eh}{\left|eh\right|}\right|\\ \mathbf{elif}\;ew \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 7.5e-155)
   (fabs (/ (* eh eh) (fabs eh)))
   (if (<= ew 4e+110)
     (fabs (/ eh (* ew (sqrt (/ 1.0 (* ew ew))))))
     (fabs (* ew t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = fabs(((eh * eh) / fabs(eh)));
	} else if (ew <= 4e+110) {
		tmp = fabs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = fabs((ew * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 7.5d-155) then
        tmp = abs(((eh * eh) / abs(eh)))
    else if (ew <= 4d+110) then
        tmp = abs((eh / (ew * sqrt((1.0d0 / (ew * ew))))))
    else
        tmp = abs((ew * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 7.5e-155) {
		tmp = Math.abs(((eh * eh) / Math.abs(eh)));
	} else if (ew <= 4e+110) {
		tmp = Math.abs((eh / (ew * Math.sqrt((1.0 / (ew * ew))))));
	} else {
		tmp = Math.abs((ew * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 7.5e-155:
		tmp = math.fabs(((eh * eh) / math.fabs(eh)))
	elif ew <= 4e+110:
		tmp = math.fabs((eh / (ew * math.sqrt((1.0 / (ew * ew))))))
	else:
		tmp = math.fabs((ew * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 7.5e-155)
		tmp = abs(Float64(Float64(eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs(Float64(eh / Float64(ew * sqrt(Float64(1.0 / Float64(ew * ew))))));
	else
		tmp = abs(Float64(ew * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 7.5e-155)
		tmp = abs(((eh * eh) / abs(eh)));
	elseif (ew <= 4e+110)
		tmp = abs((eh / (ew * sqrt((1.0 / (ew * ew))))));
	else
		tmp = abs((ew * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 7.5e-155], N[Abs[N[(N[(eh * eh), $MachinePrecision] / N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4e+110], N[Abs[N[(eh / N[(ew * N[Sqrt[N[(1.0 / N[(ew * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < 7.5000000000000006e-155

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

      3. lift-*.f64 21.5

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

   9. Applied rewrites 21.5%

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       3. rem-sqrt-square N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

       4. lower-fabs.f64 24.2

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   11. Applied rewrites 24.2%

       \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   if 7.5000000000000006e-155 < ew < 4.0000000000000001e110

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in eh around 0

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}}\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{1}{{ew}^{2}}}}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{{ew}^{2}}}}\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

      6. lift-*.f64 24.1

         \[\leadsto \left|\frac{eh}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}\right| \]

   9. Applied rewrites 24.1%

      \[\leadsto \left|\frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{1}{ew \cdot ew}}}}\right| \]

   if 4.0000000000000001e110 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|ew \cdot t\right| \]
   9. Step-by-step derivation

      1. lower-*.f64 18.9

         \[\leadsto \left|ew \cdot t\right| \]

   10. Applied rewrites 18.9%

       \[\leadsto \left|ew \cdot t\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 27.8% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 3.9 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{eh \cdot eh}{\left|eh\right|}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 3.9e+46) (fabs (/ (* eh eh) (fabs eh))) (fabs (* ew t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 3.9e+46) {
		tmp = fabs(((eh * eh) / fabs(eh)));
	} else {
		tmp = fabs((ew * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 3.9d+46) then
        tmp = abs(((eh * eh) / abs(eh)))
    else
        tmp = abs((ew * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 3.9e+46) {
		tmp = Math.abs(((eh * eh) / Math.abs(eh)));
	} else {
		tmp = Math.abs((ew * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 3.9e+46:
		tmp = math.fabs(((eh * eh) / math.fabs(eh)))
	else:
		tmp = math.fabs((ew * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 3.9e+46)
		tmp = abs(Float64(Float64(eh * eh) / abs(eh)));
	else
		tmp = abs(Float64(ew * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 3.9e+46)
		tmp = abs(((eh * eh) / abs(eh)));
	else
		tmp = abs((ew * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 3.9e+46], N[Abs[N[(N[(eh * eh), $MachinePrecision] / N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 3.89999999999999995e46

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-atan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \color{blue}{\tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh}{ew}}}{\tan t}\right)\right| \]

      5. lift-tan.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\color{blue}{\tan t}}\right)\right| \]

      6. sin-atan N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      8. associate-/l/ N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\color{blue}{\frac{eh}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{\color{blue}{ew \cdot \tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      11. lift-tan.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \color{blue}{\tan t}}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right| \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\color{blue}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      13. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{\color{blue}{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}}\right| \]

      14. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

      15. lower-pow.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \color{blue}{{\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}}\right| \]

   3. Applied rewrites 61.0%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + {\left(\frac{eh}{ew \cdot \tan t}\right)}^{2}}}}\right| \]

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{{ew}^{2}}}}\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

      10. lower-*.f64 11.3

          \[\leadsto \left|\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{eh \cdot eh}{ew \cdot \sqrt{\frac{eh \cdot eh}{ew \cdot ew}}}}\right| \]

   7. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{{eh}^{2}}}\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

      3. lift-*.f64 21.5

         \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

   9. Applied rewrites 21.5%

      \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\sqrt{eh \cdot eh}}\right| \]

       3. rem-sqrt-square N/A

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

       4. lower-fabs.f64 24.2

          \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   11. Applied rewrites 24.2%

       \[\leadsto \left|\frac{eh \cdot eh}{\left|eh\right|}\right| \]

   if 3.89999999999999995e46 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

      2. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left|ew \cdot \sin t\right| \]

      2. lift-*.f64 42.7

         \[\leadsto \left|ew \cdot \sin t\right| \]

   7. Applied rewrites 42.7%

      \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|ew \cdot t\right| \]
   9. Step-by-step derivation

      1. lower-*.f64 18.9

         \[\leadsto \left|ew \cdot t\right| \]

   10. Applied rewrites 18.9%

       \[\leadsto \left|ew \cdot t\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 18.9% accurate, 47.8× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot t\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* ew t)))

C

double code(double eh, double ew, double t) {
	return fabs((ew * t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((ew * t))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew * t));
}

Python

def code(eh, ew, t):
	return math.fabs((ew * t))

Julia

function code(eh, ew, t)
	return abs(Float64(ew * t))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew * t));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

2. Taylor expanded in eh around 0

   \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) + ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left|ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right) + \color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]

   2. associate-*r* N/A

      \[\leadsto \left|\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right) \cdot \sin t + \color{blue}{eh} \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]

   3. lower-fma.f64 N/A

      \[\leadsto \left|\mathsf{fma}\left(ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right), \color{blue}{\sin t}, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right)\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{1 + {\left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)}^{2}}} \cdot ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{\sin t \cdot ew}\right)\right)}\right| \]

5. Taylor expanded in eh around 0

   \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left|ew \cdot \sin t\right| \]

   2. lift-*.f64 42.7

      \[\leadsto \left|ew \cdot \sin t\right| \]

7. Applied rewrites 42.7%

   \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

8. Taylor expanded in t around 0

   \[\leadsto \left|ew \cdot t\right| \]
9. Step-by-step derivation

   1. lower-*.f64 18.9

      \[\leadsto \left|ew \cdot t\right| \]

10. Applied rewrites 18.9%

    \[\leadsto \left|ew \cdot t\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))