Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	return fabs((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(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 * tan(t)) / -ew))
    code = abs((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-tan.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* eh (sin t)) (sin (atan (/ (* eh (tan t)) (- ew)))))
   (*
    (* ew (cos t))
    (/ 1.0 (sqrt (+ 1.0 (pow (* (- eh) (/ (tan t) ew)) 2.0))))))))

C

double code(double eh, double ew, double t) {
	return fabs((((eh * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))) - ((ew * cos(t)) * (1.0 / sqrt((1.0 + pow((-eh * (tan(t) / ew)), 2.0)))))));
}

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((((eh * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))) - ((ew * cos(t)) * (1.0d0 / sqrt((1.0d0 + ((-eh * (tan(t) / ew)) ** 2.0d0)))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((eh * Math.sin(t)) * Math.sin(Math.atan(((eh * Math.tan(t)) / -ew)))) - ((ew * Math.cos(t)) * (1.0 / Math.sqrt((1.0 + Math.pow((-eh * (Math.tan(t) / ew)), 2.0)))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((eh * math.sin(t)) * math.sin(math.atan(((eh * math.tan(t)) / -ew)))) - ((ew * math.cos(t)) * (1.0 / math.sqrt((1.0 + math.pow((-eh * (math.tan(t) / ew)), 2.0)))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * tan(t)) / Float64(-ew))))) - Float64(Float64(ew * cos(t)) * Float64(1.0 / sqrt(Float64(1.0 + (Float64(Float64(-eh) * Float64(tan(t) / ew)) ^ 2.0)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))) - ((ew * cos(t)) * (1.0 / sqrt((1.0 + ((-eh * (tan(t) / ew)) ^ 2.0)))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 5: 99.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   ew
   (* (cos t) (cos (atan (* (- eh) (/ (tan t) ew)))))
   (* (* (sin t) (- eh)) (tanh (* (/ (- eh) ew) (tan t)))))))

C

double code(double eh, double ew, double t) {
	return fabs(fma(ew, (cos(t) * cos(atan((-eh * (tan(t) / ew))))), ((sin(t) * -eh) * tanh(((-eh / ew) * tan(t))))));
}

Julia

function code(eh, ew, t)
	return abs(fma(ew, Float64(cos(t) * cos(atan(Float64(Float64(-eh) * Float64(tan(t) / ew))))), Float64(Float64(sin(t) * Float64(-eh)) * tanh(Float64(Float64(Float64(-eh) / ew) * tan(t))))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Tanh[N[(N[((-eh) / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. times-frac N/A

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

   2. tan-quot N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-*.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Final simplification 99.2%

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

Alternative 6: 98.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[t], $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Tanh[N[ArcSinh[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in eh around 0

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

   1. cos-atan-rev N/A

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

   2. lift-cos.f64 98.6

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

9. Applied rewrites 98.6%

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

10. Final simplification 98.6%

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

Alternative 7: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{t}{ew}\\ \mathbf{if}\;eh \leq -1.12 \cdot 10^{-87} \lor \neg \left(eh \leq 8.8 \cdot 10^{-130}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \cos t \cdot \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}}, \left(\sin t \cdot \left(-eh\right)\right) \cdot \tanh \left(\frac{-eh}{ew} \cdot \tan t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ t ew))))
   (if (or (<= eh -1.12e-87) (not (<= eh 8.8e-130)))
     (fabs
      (fma
       ew
       (* (cos t) (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))))
       (* (* (sin t) (- eh)) (tanh (* (/ (- eh) ew) (tan t))))))
     (fabs (* ew (cos t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (t / ew);
	double tmp;
	if ((eh <= -1.12e-87) || !(eh <= 8.8e-130)) {
		tmp = fabs(fma(ew, (cos(t) * (1.0 / sqrt((1.0 + (t_1 * t_1))))), ((sin(t) * -eh) * tanh(((-eh / ew) * tan(t))))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(t / ew))
	tmp = 0.0
	if ((eh <= -1.12e-87) || !(eh <= 8.8e-130))
		tmp = abs(fma(ew, Float64(cos(t) * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1))))), Float64(Float64(sin(t) * Float64(-eh)) * tanh(Float64(Float64(Float64(-eh) / ew) * tan(t))))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(t / ew), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -1.12e-87], N[Not[LessEqual[eh, 8.8e-130]], $MachinePrecision]], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Tanh[N[(N[((-eh) / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.1199999999999999e-87 or 8.7999999999999995e-130 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. times-frac N/A

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

      2. tan-quot N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 90.4

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

   10. Applied rewrites 90.4%

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

       1. lift-cos.f64 N/A

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

       2. lift-atan.f64 N/A

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

       3. cos-atan N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-*.f64 90.1

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

   12. Applied rewrites 90.1%

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

   if -1.1199999999999999e-87 < eh < 8.7999999999999995e-130

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 94.6

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

   9. Applied rewrites 94.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.12 \cdot 10^{-87} \lor \neg \left(eh \leq 8.8 \cdot 10^{-130}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \cos t \cdot \frac{1}{\sqrt{1 + \left(eh \cdot \frac{t}{ew}\right) \cdot \left(eh \cdot \frac{t}{ew}\right)}}, \left(\sin t \cdot \left(-eh\right)\right) \cdot \tanh \left(\frac{-eh}{ew} \cdot \tan t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -8.2 \cdot 10^{+89} \lor \neg \left(eh \leq 1.8 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -8.2e+89) (not (<= eh 1.8e-21)))
   (fabs (* (- eh) (* (tanh (asinh (* (/ (- eh) ew) (tan t)))) (sin t))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8.2e+89) || !(eh <= 1.8e-21)) {
		tmp = fabs((-eh * (tanh(asinh(((-eh / ew) * tan(t)))) * sin(t))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -8.2e+89) or not (eh <= 1.8e-21):
		tmp = math.fabs((-eh * (math.tanh(math.asinh(((-eh / ew) * math.tan(t)))) * math.sin(t))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -8.2e+89) || !(eh <= 1.8e-21))
		tmp = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(Float64(Float64(-eh) / ew) * tan(t)))) * sin(t))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -8.2e+89) || ~((eh <= 1.8e-21)))
		tmp = abs((-eh * (tanh(asinh(((-eh / ew) * tan(t)))) * sin(t))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -8.2e+89], N[Not[LessEqual[eh, 1.8e-21]], $MachinePrecision]], N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[N[(N[((-eh) / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.1999999999999997e89 or 1.79999999999999995e-21 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 72.7%

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

   if -8.1999999999999997e89 < eh < 1.79999999999999995e-21

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 84.2

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

   9. Applied rewrites 84.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.2 \cdot 10^{+89} \lor \neg \left(eh \leq 1.8 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew} \cdot t\\ \mathbf{if}\;t \leq -0.03 \lor \neg \left(t \leq 0.042\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh, \tanh \sinh^{-1} \left(-\frac{eh \cdot t}{ew}\right) \cdot t, \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}} \cdot ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ eh ew) t)))
   (if (or (<= t -0.03) (not (<= t 0.042)))
     (fabs (* ew (cos t)))
     (fabs
      (fma
       (- eh)
       (* (tanh (asinh (- (/ (* eh t) ew)))) t)
       (* (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))) ew))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) * t;
	double tmp;
	if ((t <= -0.03) || !(t <= 0.042)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(fma(-eh, (tanh(asinh(-((eh * t) / ew))) * t), ((1.0 / sqrt((1.0 + (t_1 * t_1)))) * ew)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) * t)
	tmp = 0.0
	if ((t <= -0.03) || !(t <= 0.042))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(fma(Float64(-eh), Float64(tanh(asinh(Float64(-Float64(Float64(eh * t) / ew)))) * t), Float64(Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))) * ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t, -0.03], N[Not[LessEqual[t, 0.042]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[(-N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] + N[(N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.029999999999999999 or 0.0420000000000000026 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 51.6

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

   9. Applied rewrites 51.6%

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

   if -0.029999999999999999 < t < 0.0420000000000000026

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 98.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 98.5

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

   10. Applied rewrites 98.5%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 98.5

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

   13. Applied rewrites 98.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.03 \lor \neg \left(t \leq 0.042\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh, \tanh \sinh^{-1} \left(-\frac{eh \cdot t}{ew}\right) \cdot t, \frac{1}{\sqrt{1 + \left(\frac{eh}{ew} \cdot t\right) \cdot \left(\frac{eh}{ew} \cdot t\right)}} \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew} \cdot t\\ \mathbf{if}\;t \leq -0.03 \lor \neg \left(t \leq 0.042\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh, \tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot t, \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}} \cdot ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ eh ew) t)))
   (if (or (<= t -0.03) (not (<= t 0.042)))
     (fabs (* ew (cos t)))
     (fabs
      (fma
       (- eh)
       (* (tanh (/ (* (- eh) t) ew)) t)
       (* (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))) ew))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) * t;
	double tmp;
	if ((t <= -0.03) || !(t <= 0.042)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(fma(-eh, (tanh(((-eh * t) / ew)) * t), ((1.0 / sqrt((1.0 + (t_1 * t_1)))) * ew)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) * t)
	tmp = 0.0
	if ((t <= -0.03) || !(t <= 0.042))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(fma(Float64(-eh), Float64(tanh(Float64(Float64(Float64(-eh) * t) / ew)) * t), Float64(Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))) * ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t, -0.03], N[Not[LessEqual[t, 0.042]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-eh) * N[(N[Tanh[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] + N[(N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.029999999999999999 or 0.0420000000000000026 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 51.6

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

   9. Applied rewrites 51.6%

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

   if -0.029999999999999999 < t < 0.0420000000000000026

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 98.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 98.5

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

   10. Applied rewrites 98.5%

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

   11. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 98.0

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

   13. Applied rewrites 98.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.03 \lor \neg \left(t \leq 0.042\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh, \tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot t, \frac{1}{\sqrt{1 + \left(\frac{eh}{ew} \cdot t\right) \cdot \left(\frac{eh}{ew} \cdot t\right)}} \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \cos t\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* ew (cos t))))

C

double code(double eh, double ew, double t) {
	return fabs((ew * cos(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 * cos(t)))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew * Math.cos(t)));
}

Python

def code(eh, ew, t):
	return math.fabs((ew * math.cos(t)))

Julia

function code(eh, ew, t)
	return abs(Float64(ew * cos(t)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew * cos(t)));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in eh around 0

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

   1. cos-atan-rev N/A

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

   2. lower-*.f64 N/A

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

   3. lift-cos.f64 63.0

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

9. Applied rewrites 63.0%

   \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
10. Add Preprocessing

Alternative 12: 41.4% accurate, 287.3× speedup

Math

\[\begin{array}{l} \\ \left|ew\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs ew))

C

double code(double eh, double ew, double t) {
	return fabs(ew);
}

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)
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(ew);
}

Python

def code(eh, ew, t):
	return math.fabs(ew)

Julia

function code(eh, ew, t)
	return abs(ew)
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(ew);
end

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

   \[\leadsto \left|\color{blue}{ew}\right| \]
8. Step-by-step derivation

   1. cos-atan-rev 43.4

      \[\leadsto \left|ew\right| \]

9. Applied rewrites 43.4%

   \[\leadsto \left|\color{blue}{ew}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025035 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))