Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.6s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

Python

def code(eh, ew, t):
	t_1 = -eh * (math.tan(t) / ew)
	return math.fabs((((math.sin(t) * eh) * math.tanh(math.asinh(t_1))) - ((math.cos(t) * ew) * (1.0 / math.sqrt((1.0 + math.pow(t_1, 2.0)))))))

Julia

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

MATLAB

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

Wolfram

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

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

3. Applied rewrites 99.8%

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

Alternative 2: 97.5% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (pow
  (sqrt
   (fabs
    (- (* (* (sin t) eh) (tanh (asinh (* (- eh) (/ t ew))))) (* ew (cos t)))))
  2.0))

C

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

Python

def code(eh, ew, t):
	return math.pow(math.sqrt(math.fabs((((math.sin(t) * eh) * math.tanh(math.asinh((-eh * (t / ew))))) - (ew * math.cos(t))))), 2.0)

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = sqrt(abs((((sin(t) * eh) * tanh(asinh((-eh * (t / ew))))) - (ew * cos(t))))) ^ 2.0;
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.2%

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

4. Taylor expanded in t around 0

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

   1. lower-/.f64 98.4

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

6. Applied rewrites 98.4%

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

7. Taylor expanded in t around 0

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

   1. lower-/.f64 89.5

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

9. Applied rewrites 89.5%

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

10. Taylor expanded in eh around 0

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

    1. lift-cos.f64 N/A

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

    2. lift-*.f64 97.5

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

12. Applied rewrites 97.5%

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

Alternative 3: 63.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 8.5 \cdot 10^{-109}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 8.5e-109)
   (fabs (* (- eh) (* (tanh (asinh (- (* (/ eh ew) (tan t))))) (sin t))))
   (fabs (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 8.5e-109) {
		tmp = fabs((-eh * (tanh(asinh(-((eh / ew) * tan(t)))) * sin(t))));
	} else {
		tmp = fabs((cos(t) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 8.5e-109:
		tmp = math.fabs((-eh * (math.tanh(math.asinh(-((eh / ew) * math.tan(t)))) * math.sin(t))))
	else:
		tmp = math.fabs((math.cos(t) * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 8.5e-109)
		tmp = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * tan(t))))) * sin(t))));
	else
		tmp = abs(Float64(cos(t) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 8.5e-109)
		tmp = abs((-eh * (tanh(asinh(-((eh / ew) * tan(t)))) * sin(t))));
	else
		tmp = abs((cos(t) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 8.5e-109], N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 8.50000000000000005e-109

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 41.2%

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

   if 8.50000000000000005e-109 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in eh around 0

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

      1. lift-cos.f64 62.0

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

   7. Applied rewrites 62.0%

      \[\leadsto \left|\cos t \cdot ew\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot t\\ \mathbf{if}\;t \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{t\_1 \cdot t\_1}{ew \cdot ew}}} - \left(eh \cdot t\right) \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) t)))
   (if (<= t 6.2e-9)
     (fabs
      (-
       (* (* ew (cos t)) (/ 1.0 (sqrt (+ 1.0 (/ (* t_1 t_1) (* ew ew))))))
       (* (* eh t) (tanh (asinh (/ t_1 ew))))))
     (fabs (* (cos t) ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * t;
	double tmp;
	if (t <= 6.2e-9) {
		tmp = fabs((((ew * cos(t)) * (1.0 / sqrt((1.0 + ((t_1 * t_1) / (ew * ew)))))) - ((eh * t) * tanh(asinh((t_1 / ew))))));
	} else {
		tmp = fabs((cos(t) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -eh * t
	tmp = 0
	if t <= 6.2e-9:
		tmp = math.fabs((((ew * math.cos(t)) * (1.0 / math.sqrt((1.0 + ((t_1 * t_1) / (ew * ew)))))) - ((eh * t) * math.tanh(math.asinh((t_1 / ew))))))
	else:
		tmp = math.fabs((math.cos(t) * ew))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * t)
	tmp = 0.0
	if (t <= 6.2e-9)
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) * Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(t_1 * t_1) / Float64(ew * ew)))))) - Float64(Float64(eh * t) * tanh(asinh(Float64(t_1 / ew))))));
	else
		tmp = abs(Float64(cos(t) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -eh * t;
	tmp = 0.0;
	if (t <= 6.2e-9)
		tmp = abs((((ew * cos(t)) * (1.0 / sqrt((1.0 + ((t_1 * t_1) / (ew * ew)))))) - ((eh * t) * tanh(asinh((t_1 / ew))))));
	else
		tmp = abs((cos(t) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * t), $MachinePrecision]}, If[LessEqual[t, 6.2e-9], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(ew * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * t), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(t$95$1 / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 6.2000000000000001e-9

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 59.4%

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

   14. Applied rewrites 58.4%

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

   if 6.2000000000000001e-9 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in eh around 0

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

      1. lift-cos.f64 62.0

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

   7. Applied rewrites 62.0%

      \[\leadsto \left|\cos t \cdot ew\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 58.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-eh\right) \cdot t}{ew}\\ t_2 := \sqrt{1 + t\_1 \cdot t\_1}\\ \mathbf{if}\;t \leq 0.027:\\ \;\;\;\;\left|\left(ew \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)\right)\right) \cdot \frac{1}{t\_2} - \left(eh \cdot t\right) \cdot \frac{t\_1}{t\_2}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* (- eh) t) ew)) (t_2 (sqrt (+ 1.0 (* t_1 t_1)))))
   (if (<= t 0.027)
     (fabs
      (-
       (*
        (* ew (+ 1.0 (* (* t t) (- (* 0.041666666666666664 (* t t)) 0.5))))
        (/ 1.0 t_2))
       (* (* eh t) (/ t_1 t_2))))
     (fabs (* (cos t) ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (-eh * t) / ew;
	double t_2 = sqrt((1.0 + (t_1 * t_1)));
	double tmp;
	if (t <= 0.027) {
		tmp = fabs((((ew * (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5)))) * (1.0 / t_2)) - ((eh * t) * (t_1 / t_2))));
	} else {
		tmp = fabs((cos(t) * ew));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-eh * t) / ew
    t_2 = sqrt((1.0d0 + (t_1 * t_1)))
    if (t <= 0.027d0) then
        tmp = abs((((ew * (1.0d0 + ((t * t) * ((0.041666666666666664d0 * (t * t)) - 0.5d0)))) * (1.0d0 / t_2)) - ((eh * t) * (t_1 / t_2))))
    else
        tmp = abs((cos(t) * ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (-eh * t) / ew;
	double t_2 = Math.sqrt((1.0 + (t_1 * t_1)));
	double tmp;
	if (t <= 0.027) {
		tmp = Math.abs((((ew * (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5)))) * (1.0 / t_2)) - ((eh * t) * (t_1 / t_2))));
	} else {
		tmp = Math.abs((Math.cos(t) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (-eh * t) / ew
	t_2 = math.sqrt((1.0 + (t_1 * t_1)))
	tmp = 0
	if t <= 0.027:
		tmp = math.fabs((((ew * (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5)))) * (1.0 / t_2)) - ((eh * t) * (t_1 / t_2))))
	else:
		tmp = math.fabs((math.cos(t) * ew))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(Float64(-eh) * t) / ew)
	t_2 = sqrt(Float64(1.0 + Float64(t_1 * t_1)))
	tmp = 0.0
	if (t <= 0.027)
		tmp = abs(Float64(Float64(Float64(ew * Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(0.041666666666666664 * Float64(t * t)) - 0.5)))) * Float64(1.0 / t_2)) - Float64(Float64(eh * t) * Float64(t_1 / t_2))));
	else
		tmp = abs(Float64(cos(t) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (-eh * t) / ew;
	t_2 = sqrt((1.0 + (t_1 * t_1)));
	tmp = 0.0;
	if (t <= 0.027)
		tmp = abs((((ew * (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5)))) * (1.0 / t_2)) - ((eh * t) * (t_1 / t_2))));
	else
		tmp = abs((cos(t) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.027], N[Abs[N[(N[(N[(ew * N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(t * t), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * t), $MachinePrecision] * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0269999999999999997

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 59.4%

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

   13. Taylor expanded in t around 0

       \[\leadsto \left|\left(ew \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)\right)}\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]
   14. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left|\left(ew \cdot \left(1 + \color{blue}{{t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \left(1 + {t}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)}\right)\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

       3. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {t}^{2}} - \frac{1}{2}\right)\right)\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

       4. lift-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {t}^{2}} - \frac{1}{2}\right)\right)\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

       5. lower--.f64 N/A

          \[\leadsto \left|\left(ew \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{24} \cdot {t}^{2} - \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

       6. lower-*.f64 N/A

          \[\leadsto \left|\left(ew \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)\right)\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

       7. pow2 N/A

          \[\leadsto \left|\left(ew \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{24} \cdot \left(t \cdot t\right) - \frac{1}{2}\right)\right)\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

       8. lift-*.f64 45.4

          \[\leadsto \left|\left(ew \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)\right)\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

   15. Applied rewrites 45.4%

       \[\leadsto \left|\left(ew \cdot \color{blue}{\left(1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)\right)}\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}} - \left(eh \cdot t\right) \cdot \frac{\frac{\left(-eh\right) \cdot t}{ew}}{\sqrt{1 + \frac{\left(-eh\right) \cdot t}{ew} \cdot \frac{\left(-eh\right) \cdot t}{ew}}}\right| \]

   if 0.0269999999999999997 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in eh around 0

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

      1. lift-cos.f64 62.0

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

   7. Applied rewrites 62.0%

      \[\leadsto \left|\cos t \cdot ew\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 57.6% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in ew around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 92.1%

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

5. Taylor expanded in eh around 0

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

   1. lift-cos.f64 62.0

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

7. Applied rewrites 62.0%

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

Alternative 7: 42.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 1.5 \cdot 10^{-165}:\\ \;\;\;\;\left|\frac{\left(eh \cdot eh\right) \cdot t}{\sqrt{eh \cdot eh}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 1.5e-165) (fabs (/ (* (* eh eh) t) (sqrt (* eh eh)))) (fabs ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 1.5e-165) {
		tmp = fabs((((eh * eh) * t) / sqrt((eh * eh))));
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 1.5d-165) then
        tmp = abs((((eh * eh) * t) / sqrt((eh * eh))))
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 1.5e-165) {
		tmp = Math.abs((((eh * eh) * t) / Math.sqrt((eh * eh))));
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 1.5e-165:
		tmp = math.fabs((((eh * eh) * t) / math.sqrt((eh * eh))))
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 1.5e-165)
		tmp = abs(Float64(Float64(Float64(eh * eh) * t) / sqrt(Float64(eh * eh))));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 1.5e-165)
		tmp = abs((((eh * eh) * t) / sqrt((eh * eh))));
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 1.5e-165], N[Abs[N[(N[(N[(eh * eh), $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(eh * eh), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 1.49999999999999989e-165

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in ew around 0

       \[\leadsto \left|\color{blue}{\frac{{eh}^{2} \cdot {\sin t}^{2}}{\cos t \cdot \sqrt{\frac{{eh}^{2} \cdot {\sin t}^{2}}{{\cos t}^{2}}}}}\right| \]
   11. Step-by-step derivation

       1. cos-atan-rev N/A

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

       2. sin-atan-rev N/A

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

   12. Applied rewrites 24.5%

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

   13. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 8.7

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

   15. Applied rewrites 8.7%

       \[\leadsto \left|\frac{\left(eh \cdot eh\right) \cdot t}{\color{blue}{\sqrt{eh \cdot eh}}}\right| \]

   if 1.49999999999999989e-165 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \left|ew\right| \]
   6. Step-by-step derivation

   7. Applied rewrites 42.6%

      \[\leadsto \left|ew\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 24.9% accurate, 112.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 42.1%

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

5. Taylor expanded in eh around 0

   \[\leadsto \left|ew\right| \]
6. Step-by-step derivation

7. Applied rewrites 42.6%

   \[\leadsto \left|ew\right| \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))