Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.3s

Alternatives: 12

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   (* (tanh (asinh (/ (/ eh (tan t)) ew))) (cos t))
   eh
   (* (/ (sin t) (hypot -1.0 (/ eh (* ew t)))) ew))))

C

double code(double eh, double ew, double t) {
	return fabs(fma((tanh(asinh(((eh / tan(t)) / ew))) * cos(t)), eh, ((sin(t) / hypot(-1.0, (eh / (ew * t)))) * ew)));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(tanh(asinh(Float64(Float64(eh / tan(t)) / ew))) * cos(t)), eh, Float64(Float64(sin(t) / hypot(-1.0, Float64(eh / Float64(ew * t)))) * ew)))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[-1.0 ^ 2 + N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 98.1

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

7. Applied rewrites 98.1%

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

Alternative 4: 77.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.02 \cdot 10^{-72} \lor \neg \left(eh \leq 1.22 \cdot 10^{+64}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \frac{\frac{eh}{ew} \cdot eh}{\tan t} \cdot \cos t\right)}{\mathsf{hypot}\left(-1, \frac{eh}{ew \cdot t}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.02e-72) (not (<= eh 1.22e+64)))
   (fabs (* eh (cos t)))
   (fabs
    (/
     (fma (sin t) ew (* (/ (* (/ eh ew) eh) (tan t)) (cos t)))
     (hypot -1.0 (/ eh (* ew t)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.02e-72) || !(eh <= 1.22e+64)) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((fma(sin(t), ew, ((((eh / ew) * eh) / tan(t)) * cos(t))) / hypot(-1.0, (eh / (ew * t)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.02e-72) || !(eh <= 1.22e+64))
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(fma(sin(t), ew, Float64(Float64(Float64(Float64(eh / ew) * eh) / tan(t)) * cos(t))) / hypot(-1.0, Float64(eh / Float64(ew * t)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.02e-72], N[Not[LessEqual[eh, 1.22e+64]], $MachinePrecision]], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew + N[(N[(N[(N[(eh / ew), $MachinePrecision] * eh), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[-1.0 ^ 2 + N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.02e-72 or 1.21999999999999994e64 < eh

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -1.02e-72 < eh < 1.21999999999999994e64

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.9

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

   7. Applied rewrites 79.9%

      \[\leadsto \left|\frac{\mathsf{fma}\left(\sin t, ew, \frac{\frac{eh}{ew} \cdot eh}{\tan t} \cdot \cos t\right)}{\mathsf{hypot}\left(-1, \color{blue}{\frac{eh}{ew \cdot t}}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.02 \cdot 10^{-72} \lor \neg \left(eh \leq 1.22 \cdot 10^{+64}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \frac{\frac{eh}{ew} \cdot eh}{\tan t} \cdot \cos t\right)}{\mathsf{hypot}\left(-1, \frac{eh}{ew \cdot t}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;eh \leq -4.1 \cdot 10^{-74}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 3.5 \cdot 10^{-85}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, eh \cdot \left(eh \cdot \frac{0.5 \cdot {\cos t}^{2}}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \cos t, \left(ew \cdot ew\right) \cdot \left(0.5 \cdot \frac{{\sin t}^{2}}{t\_1}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 5.9 \cdot 10^{+63}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{ew}^{-1}}{t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))) (t_2 (fabs t_1)))
   (if (<= eh -4.1e-74)
     t_2
     (if (<= eh 3.5e-85)
       (fabs
        (fma
         ew
         (sin t)
         (* eh (* eh (/ (* 0.5 (pow (cos t) 2.0)) (* ew (sin t)))))))
       (if (<= eh 1.25e-27)
         (fabs
          (fma eh (cos t) (* (* ew ew) (* 0.5 (/ (pow (sin t) 2.0) t_1)))))
         (if (<= eh 5.9e+63)
           (fabs (fma ew (sin t) (* (* eh eh) (* 0.5 (/ (pow ew -1.0) t)))))
           t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = fabs(t_1);
	double tmp;
	if (eh <= -4.1e-74) {
		tmp = t_2;
	} else if (eh <= 3.5e-85) {
		tmp = fabs(fma(ew, sin(t), (eh * (eh * ((0.5 * pow(cos(t), 2.0)) / (ew * sin(t)))))));
	} else if (eh <= 1.25e-27) {
		tmp = fabs(fma(eh, cos(t), ((ew * ew) * (0.5 * (pow(sin(t), 2.0) / t_1)))));
	} else if (eh <= 5.9e+63) {
		tmp = fabs(fma(ew, sin(t), ((eh * eh) * (0.5 * (pow(ew, -1.0) / t)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = abs(t_1)
	tmp = 0.0
	if (eh <= -4.1e-74)
		tmp = t_2;
	elseif (eh <= 3.5e-85)
		tmp = abs(fma(ew, sin(t), Float64(eh * Float64(eh * Float64(Float64(0.5 * (cos(t) ^ 2.0)) / Float64(ew * sin(t)))))));
	elseif (eh <= 1.25e-27)
		tmp = abs(fma(eh, cos(t), Float64(Float64(ew * ew) * Float64(0.5 * Float64((sin(t) ^ 2.0) / t_1)))));
	elseif (eh <= 5.9e+63)
		tmp = abs(fma(ew, sin(t), Float64(Float64(eh * eh) * Float64(0.5 * Float64((ew ^ -1.0) / t)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[eh, -4.1e-74], t$95$2, If[LessEqual[eh, 3.5e-85], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(eh * N[(eh * N[(N[(0.5 * N[Power[N[Cos[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 1.25e-27], N[Abs[N[(eh * N[Cos[t], $MachinePrecision] + N[(N[(ew * ew), $MachinePrecision] * N[(0.5 * N[(N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 5.9e+63], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(N[(eh * eh), $MachinePrecision] * N[(0.5 * N[(N[Power[ew, -1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if eh < -4.10000000000000032e-74 or 5.90000000000000029e63 < eh

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -4.10000000000000032e-74 < eh < 3.49999999999999978e-85

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in eh around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 80.3

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 80.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 81.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, eh \cdot \left(eh \cdot \frac{0.5 \cdot {\cos t}^{2}}{ew \cdot \sin t}\right)\right)\right| \]

   if 3.49999999999999978e-85 < eh < 1.25e-27

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in ew around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 84.9

          \[\leadsto \left|\mathsf{fma}\left(eh, \cos t, \left(ew \cdot ew\right) \cdot \left(0.5 \cdot \frac{{\sin t}^{2}}{eh \cdot \color{blue}{\cos t}}\right)\right)\right| \]

   7. Applied rewrites 84.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh, \cos t, \left(ew \cdot ew\right) \cdot \left(0.5 \cdot \frac{{\sin t}^{2}}{eh \cdot \cos t}\right)\right)}\right| \]

   if 1.25e-27 < eh < 5.90000000000000029e63

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in eh around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 69.9

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 69.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 70.3%

       \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{\frac{1}{ew}}{t}\right)\right)\right| \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.1 \cdot 10^{-74}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{elif}\;eh \leq 3.5 \cdot 10^{-85}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, eh \cdot \left(eh \cdot \frac{0.5 \cdot {\cos t}^{2}}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \cos t, \left(ew \cdot ew\right) \cdot \left(0.5 \cdot \frac{{\sin t}^{2}}{eh \cdot \cos t}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 5.9 \cdot 10^{+63}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{ew}^{-1}}{t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := \left|t\_1\right|\\ t_3 := \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{ew}^{-1}}{t}\right)\right)\right|\\ \mathbf{if}\;eh \leq -4.1 \cdot 10^{-74}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 3.5 \cdot 10^{-85}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \cos t, \left(ew \cdot ew\right) \cdot \left(0.5 \cdot \frac{{\sin t}^{2}}{t\_1}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 5.9 \cdot 10^{+63}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t)))
        (t_2 (fabs t_1))
        (t_3
         (fabs (fma ew (sin t) (* (* eh eh) (* 0.5 (/ (pow ew -1.0) t)))))))
   (if (<= eh -4.1e-74)
     t_2
     (if (<= eh 3.5e-85)
       t_3
       (if (<= eh 1.25e-27)
         (fabs
          (fma eh (cos t) (* (* ew ew) (* 0.5 (/ (pow (sin t) 2.0) t_1)))))
         (if (<= eh 5.9e+63) t_3 t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = fabs(t_1);
	double t_3 = fabs(fma(ew, sin(t), ((eh * eh) * (0.5 * (pow(ew, -1.0) / t)))));
	double tmp;
	if (eh <= -4.1e-74) {
		tmp = t_2;
	} else if (eh <= 3.5e-85) {
		tmp = t_3;
	} else if (eh <= 1.25e-27) {
		tmp = fabs(fma(eh, cos(t), ((ew * ew) * (0.5 * (pow(sin(t), 2.0) / t_1)))));
	} else if (eh <= 5.9e+63) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = abs(t_1)
	t_3 = abs(fma(ew, sin(t), Float64(Float64(eh * eh) * Float64(0.5 * Float64((ew ^ -1.0) / t)))))
	tmp = 0.0
	if (eh <= -4.1e-74)
		tmp = t_2;
	elseif (eh <= 3.5e-85)
		tmp = t_3;
	elseif (eh <= 1.25e-27)
		tmp = abs(fma(eh, cos(t), Float64(Float64(ew * ew) * Float64(0.5 * Float64((sin(t) ^ 2.0) / t_1)))));
	elseif (eh <= 5.9e+63)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(N[(eh * eh), $MachinePrecision] * N[(0.5 * N[(N[Power[ew, -1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -4.1e-74], t$95$2, If[LessEqual[eh, 3.5e-85], t$95$3, If[LessEqual[eh, 1.25e-27], N[Abs[N[(eh * N[Cos[t], $MachinePrecision] + N[(N[(ew * ew), $MachinePrecision] * N[(0.5 * N[(N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 5.9e+63], t$95$3, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -4.10000000000000032e-74 or 5.90000000000000029e63 < eh

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -4.10000000000000032e-74 < eh < 3.49999999999999978e-85 or 1.25e-27 < eh < 5.90000000000000029e63

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in eh around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 78.1

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 78.1%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.2%

       \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{\frac{1}{ew}}{t}\right)\right)\right| \]

   if 3.49999999999999978e-85 < eh < 1.25e-27

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in ew around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 84.9

          \[\leadsto \left|\mathsf{fma}\left(eh, \cos t, \left(ew \cdot ew\right) \cdot \left(0.5 \cdot \frac{{\sin t}^{2}}{eh \cdot \color{blue}{\cos t}}\right)\right)\right| \]

   7. Applied rewrites 84.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh, \cos t, \left(ew \cdot ew\right) \cdot \left(0.5 \cdot \frac{{\sin t}^{2}}{eh \cdot \cos t}\right)\right)}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.1 \cdot 10^{-74}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{elif}\;eh \leq 3.5 \cdot 10^{-85}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{ew}^{-1}}{t}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \cos t, \left(ew \cdot ew\right) \cdot \left(0.5 \cdot \frac{{\sin t}^{2}}{eh \cdot \cos t}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 5.9 \cdot 10^{+63}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{ew}^{-1}}{t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \cos t\right|\\ \mathbf{if}\;eh \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq -3.3 \cdot 10^{-200}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right|\\ \mathbf{elif}\;eh \leq 2.15 \cdot 10^{-227}:\\ \;\;\;\;{\left({\left(ew \cdot \sin t\right)}^{-1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (cos t)))))
   (if (<= eh -6.8e-76)
     t_1
     (if (<= eh -3.3e-200)
       (fabs
        (/
         (fma
          (/ eh ew)
          (* eh 0.5)
          (* (* (fma -0.4166666666666667 (* (/ eh ew) eh) ew) t) t))
         t))
       (if (<= eh 2.15e-227) (pow (pow (* ew (sin t)) -1.0) -1.0) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * cos(t)));
	double tmp;
	if (eh <= -6.8e-76) {
		tmp = t_1;
	} else if (eh <= -3.3e-200) {
		tmp = fabs((fma((eh / ew), (eh * 0.5), ((fma(-0.4166666666666667, ((eh / ew) * eh), ew) * t) * t)) / t));
	} else if (eh <= 2.15e-227) {
		tmp = pow(pow((ew * sin(t)), -1.0), -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * cos(t)))
	tmp = 0.0
	if (eh <= -6.8e-76)
		tmp = t_1;
	elseif (eh <= -3.3e-200)
		tmp = abs(Float64(fma(Float64(eh / ew), Float64(eh * 0.5), Float64(Float64(fma(-0.4166666666666667, Float64(Float64(eh / ew) * eh), ew) * t) * t)) / t));
	elseif (eh <= 2.15e-227)
		tmp = (Float64(ew * sin(t)) ^ -1.0) ^ -1.0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -6.8e-76], t$95$1, If[LessEqual[eh, -3.3e-200], N[Abs[N[(N[(N[(eh / ew), $MachinePrecision] * N[(eh * 0.5), $MachinePrecision] + N[(N[(N[(-0.4166666666666667 * N[(N[(eh / ew), $MachinePrecision] * eh), $MachinePrecision] + ew), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 2.15e-227], N[Power[N[Power[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -6.7999999999999998e-76 or 2.1500000000000001e-227 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 60.2%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 74.0

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

   7. Applied rewrites 74.0%

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

   if -6.7999999999999998e-76 < eh < -3.2999999999999998e-200

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 86.4

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 86.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 50.5%

       \[\leadsto \left|\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.5, \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334, ew\right), 0.5 \cdot \frac{eh \cdot eh}{ew}\right)}{\color{blue}{t}}\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 59.6%

       \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right| \]

   if -3.2999999999999998e-200 < eh < 2.1500000000000001e-227

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 56.5%

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

   7. Taylor expanded in eh around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew \cdot \sin t}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew \cdot \sin t}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{ew \cdot \sin t}}} \]

      3. lower-sin.f64 49.6

         \[\leadsto \frac{1}{\frac{1}{ew \cdot \color{blue}{\sin t}}} \]

   9. Applied rewrites 49.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew \cdot \sin t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{elif}\;eh \leq -3.3 \cdot 10^{-200}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right|\\ \mathbf{elif}\;eh \leq 2.15 \cdot 10^{-227}:\\ \;\;\;\;{\left({\left(ew \cdot \sin t\right)}^{-1}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -4.1 \cdot 10^{-74} \lor \neg \left(eh \leq 3.5 \cdot 10^{-85} \lor \neg \left(eh \leq 1.25 \cdot 10^{-27} \lor \neg \left(eh \leq 5.9 \cdot 10^{+63}\right)\right)\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{ew}^{-1}}{t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -4.1e-74)
         (not
          (or (<= eh 3.5e-85)
              (not (or (<= eh 1.25e-27) (not (<= eh 5.9e+63)))))))
   (fabs (* eh (cos t)))
   (fabs (fma ew (sin t) (* (* eh eh) (* 0.5 (/ (pow ew -1.0) t)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -4.1e-74) || !((eh <= 3.5e-85) || !((eh <= 1.25e-27) || !(eh <= 5.9e+63)))) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs(fma(ew, sin(t), ((eh * eh) * (0.5 * (pow(ew, -1.0) / t)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -4.1e-74) || !((eh <= 3.5e-85) || !((eh <= 1.25e-27) || !(eh <= 5.9e+63))))
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(fma(ew, sin(t), Float64(Float64(eh * eh) * Float64(0.5 * Float64((ew ^ -1.0) / t)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -4.1e-74], N[Not[Or[LessEqual[eh, 3.5e-85], N[Not[Or[LessEqual[eh, 1.25e-27], N[Not[LessEqual[eh, 5.9e+63]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(N[(eh * eh), $MachinePrecision] * N[(0.5 * N[(N[Power[ew, -1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.10000000000000032e-74 or 3.49999999999999978e-85 < eh < 1.25e-27 or 5.90000000000000029e63 < eh

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 87.3

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

   7. Applied rewrites 87.3%

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

   if -4.10000000000000032e-74 < eh < 3.49999999999999978e-85 or 1.25e-27 < eh < 5.90000000000000029e63

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in eh around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 78.1

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 78.1%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.2%

       \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{\frac{1}{ew}}{t}\right)\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.1 \cdot 10^{-74} \lor \neg \left(eh \leq 3.5 \cdot 10^{-85} \lor \neg \left(eh \leq 1.25 \cdot 10^{-27} \lor \neg \left(eh \leq 5.9 \cdot 10^{+63}\right)\right)\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{ew}^{-1}}{t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{-76} \lor \neg \left(eh \leq 4 \cdot 10^{-279}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -6.8e-76) (not (<= eh 4e-279)))
   (fabs (* eh (cos t)))
   (fabs
    (/
     (fma
      (/ eh ew)
      (* eh 0.5)
      (* (* (fma -0.4166666666666667 (* (/ eh ew) eh) ew) t) t))
     t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -6.8e-76) || !(eh <= 4e-279)) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((fma((eh / ew), (eh * 0.5), ((fma(-0.4166666666666667, ((eh / ew) * eh), ew) * t) * t)) / t));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -6.8e-76) || !(eh <= 4e-279))
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(fma(Float64(eh / ew), Float64(eh * 0.5), Float64(Float64(fma(-0.4166666666666667, Float64(Float64(eh / ew) * eh), ew) * t) * t)) / t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -6.8e-76], N[Not[LessEqual[eh, 4e-279]], $MachinePrecision]], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh / ew), $MachinePrecision] * N[(eh * 0.5), $MachinePrecision] + N[(N[(N[(-0.4166666666666667 * N[(N[(eh / ew), $MachinePrecision] * eh), $MachinePrecision] + ew), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.7999999999999998e-76 or 4.00000000000000022e-279 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 71.1

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

   7. Applied rewrites 71.1%

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

   if -6.7999999999999998e-76 < eh < 4.00000000000000022e-279

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in eh around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 87.8

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 87.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 31.3%

       \[\leadsto \left|\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.5, \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334, ew\right), 0.5 \cdot \frac{eh \cdot eh}{ew}\right)}{\color{blue}{t}}\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 39.3%

       \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{-76} \lor \neg \left(eh \leq 4 \cdot 10^{-279}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.7% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.1 \cdot 10^{-75} \lor \neg \left(eh \leq 4 \cdot 10^{-279}\right):\\ \;\;\;\;\left|eh \cdot 1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.1e-75) (not (<= eh 4e-279)))
   (fabs (* eh 1.0))
   (fabs
    (/
     (fma
      (/ eh ew)
      (* eh 0.5)
      (* (* (fma -0.4166666666666667 (* (/ eh ew) eh) ew) t) t))
     t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.1e-75) || !(eh <= 4e-279)) {
		tmp = fabs((eh * 1.0));
	} else {
		tmp = fabs((fma((eh / ew), (eh * 0.5), ((fma(-0.4166666666666667, ((eh / ew) * eh), ew) * t) * t)) / t));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.1e-75) || !(eh <= 4e-279))
		tmp = abs(Float64(eh * 1.0));
	else
		tmp = abs(Float64(fma(Float64(eh / ew), Float64(eh * 0.5), Float64(Float64(fma(-0.4166666666666667, Float64(Float64(eh / ew) * eh), ew) * t) * t)) / t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.1e-75], N[Not[LessEqual[eh, 4e-279]], $MachinePrecision]], N[Abs[N[(eh * 1.0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh / ew), $MachinePrecision] * N[(eh * 0.5), $MachinePrecision] + N[(N[(N[(-0.4166666666666667 * N[(N[(eh / ew), $MachinePrecision] * eh), $MachinePrecision] + ew), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.10000000000000003e-75 or 4.00000000000000022e-279 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|eh \cdot 1\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

       \[\leadsto \left|eh \cdot 1\right| \]

   if -1.10000000000000003e-75 < eh < 4.00000000000000022e-279

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in eh around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 87.8

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 87.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 31.3%

       \[\leadsto \left|\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.5, \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334, ew\right), 0.5 \cdot \frac{eh \cdot eh}{ew}\right)}{\color{blue}{t}}\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 39.3%

       \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.1 \cdot 10^{-75} \lor \neg \left(eh \leq 4 \cdot 10^{-279}\right):\\ \;\;\;\;\left|eh \cdot 1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew}, eh \cdot 0.5, \left(\mathsf{fma}\left(-0.4166666666666667, \frac{eh}{ew} \cdot eh, ew\right) \cdot t\right) \cdot t\right)}{t}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.1% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{-76} \lor \neg \left(eh \leq -6.8 \cdot 10^{-230}\right):\\ \;\;\;\;\left|eh \cdot 1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \left(t \cdot t\right)}{t}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -6.8e-76) (not (<= eh -6.8e-230)))
   (fabs (* eh 1.0))
   (fabs (/ (* ew (* t t)) t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -6.8e-76) || !(eh <= -6.8e-230)) {
		tmp = fabs((eh * 1.0));
	} else {
		tmp = fabs(((ew * (t * t)) / t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-6.8d-76)) .or. (.not. (eh <= (-6.8d-230)))) then
        tmp = abs((eh * 1.0d0))
    else
        tmp = abs(((ew * (t * t)) / t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -6.8e-76) || !(eh <= -6.8e-230)) {
		tmp = Math.abs((eh * 1.0));
	} else {
		tmp = Math.abs(((ew * (t * t)) / t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -6.8e-76) or not (eh <= -6.8e-230):
		tmp = math.fabs((eh * 1.0))
	else:
		tmp = math.fabs(((ew * (t * t)) / t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -6.8e-76) || !(eh <= -6.8e-230))
		tmp = abs(Float64(eh * 1.0));
	else
		tmp = abs(Float64(Float64(ew * Float64(t * t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -6.8e-76) || ~((eh <= -6.8e-230)))
		tmp = abs((eh * 1.0));
	else
		tmp = abs(((ew * (t * t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -6.8e-76], N[Not[LessEqual[eh, -6.8e-230]], $MachinePrecision]], N[Abs[N[(eh * 1.0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.7999999999999998e-76 or -6.8e-230 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 66.6

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

   7. Applied rewrites 66.6%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|eh \cdot 1\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 44.3%

       \[\leadsto \left|eh \cdot 1\right| \]

   if -6.7999999999999998e-76 < eh < -6.8e-230

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in eh around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 87.3

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 87.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 44.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.5, \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334, ew\right), 0.5 \cdot \frac{eh \cdot eh}{ew}\right)}{\color{blue}{t}}\right| \]

   11. Taylor expanded in eh around 0

       \[\leadsto \left|\frac{ew \cdot {t}^{2}}{t}\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 44.3%

       \[\leadsto \left|\frac{ew \cdot \left(t \cdot t\right)}{t}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{-76} \lor \neg \left(eh \leq -6.8 \cdot 10^{-230}\right):\\ \;\;\;\;\left|eh \cdot 1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \left(t \cdot t\right)}{t}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.9% accurate, 108.8× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot 1\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* eh 1.0)))

C

double code(double eh, double ew, double t) {
	return fabs((eh * 1.0));
}

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 * 1.0d0))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((eh * 1.0));
}

Python

def code(eh, ew, t):
	return math.fabs((eh * 1.0))

Julia

function code(eh, ew, t)
	return abs(Float64(eh * 1.0))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((eh * 1.0));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh * 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. cos-atan N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. sin-atan N/A

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

4. Applied rewrites 67.8%

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

5. Taylor expanded in eh around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 61.0

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

7. Applied rewrites 61.0%

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

8. Taylor expanded in t around 0

   \[\leadsto \left|eh \cdot 1\right| \]
9. Step-by-step derivation

10. Applied rewrites 40.9%

    \[\leadsto \left|eh \cdot 1\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024357 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))