Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.5s

Alternatives: 11

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in t around 0

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

6. Applied rewrites 99.1%

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

Alternative 3: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lift-sin.f64 98.5

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

6. Applied rewrites 98.5%

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

Alternative 4: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -3 \cdot 10^{+117}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;\left|t\_1 \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{\sin t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)) (t_2 (fabs (* ew (sin t)))))
   (if (<= ew -3e+117)
     t_2
     (if (<= ew 7.5e-24)
       (fabs (* t_1 (tanh (asinh (/ t_1 (* (sin t) ew))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -3e+117) {
		tmp = t_2;
	} else if (ew <= 7.5e-24) {
		tmp = fabs((t_1 * tanh(asinh((t_1 / (sin(t) * ew))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * eh
	t_2 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -3e+117:
		tmp = t_2
	elif ew <= 7.5e-24:
		tmp = math.fabs((t_1 * math.tanh(math.asinh((t_1 / (math.sin(t) * ew))))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -3e+117)
		tmp = t_2;
	elseif (ew <= 7.5e-24)
		tmp = abs(Float64(t_1 * tanh(asinh(Float64(t_1 / Float64(sin(t) * ew))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * eh;
	t_2 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -3e+117)
		tmp = t_2;
	elseif (ew <= 7.5e-24)
		tmp = abs((t_1 * tanh(asinh((t_1 / (sin(t) * ew))))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3e+117], t$95$2, If[LessEqual[ew, 7.5e-24], N[Abs[N[(t$95$1 * N[Tanh[N[ArcSinh[N[(t$95$1 / N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3e117 or 7.50000000000000007e-24 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 65.6

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

   6. Applied rewrites 65.6%

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

   if -3e117 < ew < 7.50000000000000007e-24

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sin.f64 79.5

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

   4. Applied rewrites 79.5%

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

Alternative 5: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -0.055:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+37}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), \left(t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right) \cdot 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -0.055)
     t_1
     (if (<= t 6.2e+37)
       (fabs
        (fma
         (* (cos t) eh)
         (tanh (asinh (/ eh (* ew (tan t)))))
         (* (* t (+ ew (* -0.16666666666666666 (* ew (* t t))))) 1.0)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -0.055) {
		tmp = t_1;
	} else if (t <= 6.2e+37) {
		tmp = fabs(fma((cos(t) * eh), tanh(asinh((eh / (ew * tan(t))))), ((t * (ew + (-0.16666666666666666 * (ew * (t * t))))) * 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -0.055)
		tmp = t_1;
	elseif (t <= 6.2e+37)
		tmp = abs(fma(Float64(cos(t) * eh), tanh(asinh(Float64(eh / Float64(ew * tan(t))))), Float64(Float64(t * Float64(ew + Float64(-0.16666666666666666 * Float64(ew * Float64(t * t))))) * 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.055], t$95$1, If[LessEqual[t, 6.2e+37], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[(t * N[(ew + N[(-0.16666666666666666 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0550000000000000003 or 6.2000000000000004e37 < t

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 51.6

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

   6. Applied rewrites 51.6%

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

   if -0.0550000000000000003 < t < 6.2000000000000004e37

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 96.5

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

   6. Applied rewrites 96.5%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 95.2%

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), \left(t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right)\right) \cdot \color{blue}{1}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -3 \cdot 10^{+117}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq -4.3 \cdot 10^{-79}:\\ \;\;\;\;\left|t\_1 \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)\right|\\ \mathbf{elif}\;ew \leq 1.26 \cdot 10^{-58}:\\ \;\;\;\;\left|t\_1 \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)) (t_2 (fabs (* ew (sin t)))))
   (if (<= ew -3e+117)
     t_2
     (if (<= ew -4.3e-79)
       (fabs
        (*
         t_1
         (tanh
          (asinh
           (/
            (fma
             (* t t)
             (- (* -0.5 (/ eh ew)) (* -0.16666666666666666 (/ eh ew)))
             (/ eh ew))
            t)))))
       (if (<= ew 1.26e-58)
         (fabs (* t_1 (tanh (asinh (/ eh (* ew t))))))
         t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -3e+117) {
		tmp = t_2;
	} else if (ew <= -4.3e-79) {
		tmp = fabs((t_1 * tanh(asinh((fma((t * t), ((-0.5 * (eh / ew)) - (-0.16666666666666666 * (eh / ew))), (eh / ew)) / t)))));
	} else if (ew <= 1.26e-58) {
		tmp = fabs((t_1 * tanh(asinh((eh / (ew * t))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -3e+117)
		tmp = t_2;
	elseif (ew <= -4.3e-79)
		tmp = abs(Float64(t_1 * tanh(asinh(Float64(fma(Float64(t * t), Float64(Float64(-0.5 * Float64(eh / ew)) - Float64(-0.16666666666666666 * Float64(eh / ew))), Float64(eh / ew)) / t)))));
	elseif (ew <= 1.26e-58)
		tmp = abs(Float64(t_1 * tanh(asinh(Float64(eh / Float64(ew * t))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3e+117], t$95$2, If[LessEqual[ew, -4.3e-79], N[Abs[N[(t$95$1 * N[Tanh[N[ArcSinh[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.26e-58], N[Abs[N[(t$95$1 * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3e117 or 1.2600000000000001e-58 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 64.0

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

   6. Applied rewrites 64.0%

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

   if -3e117 < ew < -4.29999999999999982e-79

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sin.f64 60.2

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

   4. Applied rewrites 60.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 57.2

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

   7. Applied rewrites 57.2%

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

   if -4.29999999999999982e-79 < ew < 1.2600000000000001e-58

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sin.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in t around 0

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 74.9

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

   7. Applied rewrites 74.9%

      \[\leadsto \left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -8 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.26 \cdot 10^{-58}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -8e+19)
     t_1
     (if (<= ew 1.26e-58)
       (fabs (* (* (cos t) eh) (tanh (asinh (/ eh (* ew t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -8e+19) {
		tmp = t_1;
	} else if (ew <= 1.26e-58) {
		tmp = fabs(((cos(t) * eh) * tanh(asinh((eh / (ew * t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -8e+19:
		tmp = t_1
	elif ew <= 1.26e-58:
		tmp = math.fabs(((math.cos(t) * eh) * math.tanh(math.asinh((eh / (ew * t))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -8e+19)
		tmp = t_1;
	elseif (ew <= 1.26e-58)
		tmp = abs(Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(eh / Float64(ew * t))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -8e+19)
		tmp = t_1;
	elseif (ew <= 1.26e-58)
		tmp = abs(((cos(t) * eh) * tanh(asinh((eh / (ew * t))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -8e+19], t$95$1, If[LessEqual[ew, 1.26e-58], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -8e19 or 1.2600000000000001e-58 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 62.4

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

   6. Applied rewrites 62.4%

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

   if -8e19 < ew < 1.2600000000000001e-58

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sin.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in t around 0

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 72.2

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

   7. Applied rewrites 72.2%

      \[\leadsto \left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -1.6e-51)
     t_1
     (if (<= t 3.2e-89) (fabs (* (tanh (asinh (/ eh (* ew t)))) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -1.6e-51) {
		tmp = t_1;
	} else if (t <= 3.2e-89) {
		tmp = fabs((tanh(asinh((eh / (ew * t)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if t <= -1.6e-51:
		tmp = t_1
	elif t <= 3.2e-89:
		tmp = math.fabs((math.tanh(math.asinh((eh / (ew * t)))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -1.6e-51)
		tmp = t_1;
	elseif (t <= 3.2e-89)
		tmp = abs(Float64(tanh(asinh(Float64(eh / Float64(ew * t)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (t <= -1.6e-51)
		tmp = t_1;
	elseif (t <= 3.2e-89)
		tmp = abs((tanh(asinh((eh / (ew * t)))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.6e-51], t$95$1, If[LessEqual[t, 3.2e-89], N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.6e-51 or 3.19999999999999998e-89 < t

   1. Initial program 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 50.9

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

   6. Applied rewrites 50.9%

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

   if -1.6e-51 < t < 3.19999999999999998e-89

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 46.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right|\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+80}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= t -8.5e-6)
     t_1
     (if (<= t 3.2e-89)
       (fabs (* (tanh (asinh (/ eh (* ew t)))) eh))
       (if (<= t 5.8e+80) (fabs (* ew t)) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (t <= -8.5e-6) {
		tmp = t_1;
	} else if (t <= 3.2e-89) {
		tmp = fabs((tanh(asinh((eh / (ew * t)))) * eh));
	} else if (t <= 5.8e+80) {
		tmp = fabs((ew * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	tmp = 0
	if t <= -8.5e-6:
		tmp = t_1
	elif t <= 3.2e-89:
		tmp = math.fabs((math.tanh(math.asinh((eh / (ew * t)))) * eh))
	elif t <= 5.8e+80:
		tmp = math.fabs((ew * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (t <= -8.5e-6)
		tmp = t_1;
	elseif (t <= 3.2e-89)
		tmp = abs(Float64(tanh(asinh(Float64(eh / Float64(ew * t)))) * eh));
	elseif (t <= 5.8e+80)
		tmp = abs(Float64(ew * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if (t <= -8.5e-6)
		tmp = t_1;
	elseif (t <= 3.2e-89)
		tmp = abs((tanh(asinh((eh / (ew * t)))) * eh));
	elseif (t <= 5.8e+80)
		tmp = abs((ew * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e-6], t$95$1, If[LessEqual[t, 3.2e-89], N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 5.8e+80], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.4999999999999999e-6 or 5.79999999999999971e80 < t

   1. Initial program 99.6%

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

   3. Applied rewrites 50.7%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 27.3

         \[\leadsto ew \cdot \sin t \]

   6. Applied rewrites 27.3%

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

   if -8.4999999999999999e-6 < t < 3.19999999999999998e-89

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if 3.19999999999999998e-89 < t < 5.79999999999999971e80

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 5.0

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 4.9%

       \[\leadsto \left|\left(\frac{1}{\frac{eh}{ew \cdot t}} \cdot ew\right) \cdot t\right| \]

   11. Taylor expanded in eh around 0

       \[\leadsto \left|ew \cdot t\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 26.7%

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

Alternative 10: 42.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot t\right|\\ \mathbf{if}\;ew \leq -2.35 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.5 \cdot 10^{+172}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew t))))
   (if (<= ew -2.35e+168)
     t_1
     (if (<= ew 1.5e+172) (fabs (* (tanh (asinh (/ eh (* ew t)))) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * t));
	double tmp;
	if (ew <= -2.35e+168) {
		tmp = t_1;
	} else if (ew <= 1.5e+172) {
		tmp = fabs((tanh(asinh((eh / (ew * t)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * t))
	tmp = 0
	if ew <= -2.35e+168:
		tmp = t_1
	elif ew <= 1.5e+172:
		tmp = math.fabs((math.tanh(math.asinh((eh / (ew * t)))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * t))
	tmp = 0.0
	if (ew <= -2.35e+168)
		tmp = t_1;
	elseif (ew <= 1.5e+172)
		tmp = abs(Float64(tanh(asinh(Float64(eh / Float64(ew * t)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * t));
	tmp = 0.0;
	if (ew <= -2.35e+168)
		tmp = t_1;
	elseif (ew <= 1.5e+172)
		tmp = abs((tanh(asinh((eh / (ew * t)))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.35e+168], t$95$1, If[LessEqual[ew, 1.5e+172], N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.3499999999999998e168 or 1.5e172 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 6.3

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

   7. Applied rewrites 6.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 6.3%

       \[\leadsto \left|\left(\frac{1}{\frac{eh}{ew \cdot t}} \cdot ew\right) \cdot t\right| \]

   11. Taylor expanded in eh around 0

       \[\leadsto \left|ew \cdot t\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 33.8%

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

   if -2.3499999999999998e168 < ew < 1.5e172

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 45.4

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

   7. Applied rewrites 45.4%

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 18.4% accurate, 47.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((ew * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 41.3%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 4.7

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

7. Applied rewrites 4.7%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 4.6%

    \[\leadsto \left|\left(\frac{1}{\frac{eh}{ew \cdot t}} \cdot ew\right) \cdot t\right| \]

11. Taylor expanded in eh around 0

    \[\leadsto \left|ew \cdot t\right| \]
12. Step-by-step derivation

13. Applied rewrites 18.4%

    \[\leadsto \left|ew \cdot t\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(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))))))))