Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 10.9s

Alternatives: 8

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

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) * cos(atan(t_1)))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ t_2 := \tanh \sinh^{-1} t\_1\\ t_3 := \tan^{-1} t\_1\\ t_4 := \cos t\_3\\ t_5 := \left(\sin t \cdot ew\right) \cdot t\_4\\ \mathbf{if}\;\left(ew \cdot \sin t\right) \cdot t\_4 + \left(eh \cdot \cos t\right) \cdot \sin t\_3 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t\_2, t\_5\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t \cdot eh, t\_2, t\_5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t)))
        (t_2 (tanh (asinh t_1)))
        (t_3 (atan t_1))
        (t_4 (cos t_3))
        (t_5 (* (* (sin t) ew) t_4)))
   (if (<= (+ (* (* ew (sin t)) t_4) (* (* eh (cos t)) (sin t_3))) -1e-262)
     (fabs (fma eh t_2 t_5))
     (fma (* (cos t) eh) t_2 t_5))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	double t_2 = tanh(asinh(t_1));
	double t_3 = atan(t_1);
	double t_4 = cos(t_3);
	double t_5 = (sin(t) * ew) * t_4;
	double tmp;
	if ((((ew * sin(t)) * t_4) + ((eh * cos(t)) * sin(t_3))) <= -1e-262) {
		tmp = fabs(fma(eh, t_2, t_5));
	} else {
		tmp = fma((cos(t) * eh), t_2, t_5);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	t_2 = tanh(asinh(t_1))
	t_3 = atan(t_1)
	t_4 = cos(t_3)
	t_5 = Float64(Float64(sin(t) * ew) * t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(ew * sin(t)) * t_4) + Float64(Float64(eh * cos(t)) * sin(t_3))) <= -1e-262)
		tmp = abs(fma(eh, t_2, t_5));
	else
		tmp = fma(Float64(cos(t) * eh), t_2, t_5);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * t$95$4), $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-262], N[Abs[N[(eh * t$95$2 + t$95$5), $MachinePrecision]], $MachinePrecision], N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * t$95$2 + t$95$5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -1.00000000000000001e-262

   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%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 79.7%

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

   if -1.00000000000000001e-262 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 98.6%

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

Alternative 3: 86.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{1 + -0.5 \cdot \left(t \cdot t\right)}{ew} \cdot \frac{eh}{\sin t}\right)\right|\\ t_2 := \frac{\frac{eh}{ew}}{\tan t}\\ \mathbf{if}\;eh \leq -5.1 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.3 \cdot 10^{+54}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \tanh \sinh^{-1} t\_2, \left(\sin t \cdot ew\right) \cdot \cos \tan^{-1} t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (cos t) eh)
           (tanh (asinh (* (/ (+ 1.0 (* -0.5 (* t t))) ew) (/ eh (sin t))))))))
        (t_2 (/ (/ eh ew) (tan t))))
   (if (<= eh -5.1e+157)
     t_1
     (if (<= eh 2.3e+54)
       (fabs (fma eh (tanh (asinh t_2)) (* (* (sin t) ew) (cos (atan t_2)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((cos(t) * eh) * tanh(asinh((((1.0 + (-0.5 * (t * t))) / ew) * (eh / sin(t)))))));
	double t_2 = (eh / ew) / tan(t);
	double tmp;
	if (eh <= -5.1e+157) {
		tmp = t_1;
	} else if (eh <= 2.3e+54) {
		tmp = fabs(fma(eh, tanh(asinh(t_2)), ((sin(t) * ew) * cos(atan(t_2)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(Float64(Float64(1.0 + Float64(-0.5 * Float64(t * t))) / ew) * Float64(eh / sin(t)))))))
	t_2 = Float64(Float64(eh / ew) / tan(t))
	tmp = 0.0
	if (eh <= -5.1e+157)
		tmp = t_1;
	elseif (eh <= 2.3e+54)
		tmp = abs(fma(eh, tanh(asinh(t_2)), Float64(Float64(sin(t) * ew) * cos(atan(t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(N[(1.0 + N[(-0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -5.1e+157], t$95$1, If[LessEqual[eh, 2.3e+54], N[Abs[N[(eh * N[Tanh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.09999999999999999e157 or 2.29999999999999994e54 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 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. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

   if -5.09999999999999999e157 < eh < 2.29999999999999994e54

   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%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 85.6%

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

Alternative 4: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{1 + -0.5 \cdot \left(t \cdot t\right)}{ew} \cdot \frac{eh}{\sin t}\right)\right|\\ \mathbf{if}\;eh \leq -1.7 \cdot 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 10^{-83}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot ew\right) \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (cos t) eh)
           (tanh
            (asinh (* (/ (+ 1.0 (* -0.5 (* t t))) ew) (/ eh (sin t)))))))))
   (if (<= eh -1.7e-21)
     t_1
     (if (<= eh 1e-83)
       (fabs (* (* (cos (atan (/ eh (* ew t)))) ew) (sin t)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((cos(t) * eh) * tanh(asinh((((1.0 + (-0.5 * (t * t))) / ew) * (eh / sin(t)))))));
	double tmp;
	if (eh <= -1.7e-21) {
		tmp = t_1;
	} else if (eh <= 1e-83) {
		tmp = fabs(((cos(atan((eh / (ew * t)))) * ew) * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((math.cos(t) * eh) * math.tanh(math.asinh((((1.0 + (-0.5 * (t * t))) / ew) * (eh / math.sin(t)))))))
	tmp = 0
	if eh <= -1.7e-21:
		tmp = t_1
	elif eh <= 1e-83:
		tmp = math.fabs(((math.cos(math.atan((eh / (ew * t)))) * ew) * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(Float64(Float64(1.0 + Float64(-0.5 * Float64(t * t))) / ew) * Float64(eh / sin(t)))))))
	tmp = 0.0
	if (eh <= -1.7e-21)
		tmp = t_1;
	elseif (eh <= 1e-83)
		tmp = abs(Float64(Float64(cos(atan(Float64(eh / Float64(ew * t)))) * ew) * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((cos(t) * eh) * tanh(asinh((((1.0 + (-0.5 * (t * t))) / ew) * (eh / sin(t)))))));
	tmp = 0.0;
	if (eh <= -1.7e-21)
		tmp = t_1;
	elseif (eh <= 1e-83)
		tmp = abs(((cos(atan((eh / (ew * t)))) * ew) * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(N[(1.0 + N[(-0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.7e-21], t$95$1, If[LessEqual[eh, 1e-83], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.7e-21 or 1e-83 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 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. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if -1.7e-21 < eh < 1e-83

   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 67.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 67.6

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

   7. Applied rewrites 67.6%

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

Alternative 5: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{\sin t}\right)\right|\\ \mathbf{if}\;eh \leq -1.7 \cdot 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.95 \cdot 10^{-83}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot ew\right) \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (* (cos t) eh) (tanh (asinh (* (/ 1.0 ew) (/ eh (sin t)))))))))
   (if (<= eh -1.7e-21)
     t_1
     (if (<= eh 1.95e-83)
       (fabs (* (* (cos (atan (/ eh (* ew t)))) ew) (sin t)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((cos(t) * eh) * tanh(asinh(((1.0 / ew) * (eh / sin(t)))))));
	double tmp;
	if (eh <= -1.7e-21) {
		tmp = t_1;
	} else if (eh <= 1.95e-83) {
		tmp = fabs(((cos(atan((eh / (ew * t)))) * ew) * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((math.cos(t) * eh) * math.tanh(math.asinh(((1.0 / ew) * (eh / math.sin(t)))))))
	tmp = 0
	if eh <= -1.7e-21:
		tmp = t_1
	elif eh <= 1.95e-83:
		tmp = math.fabs(((math.cos(math.atan((eh / (ew * t)))) * ew) * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(Float64(1.0 / ew) * Float64(eh / sin(t)))))))
	tmp = 0.0
	if (eh <= -1.7e-21)
		tmp = t_1;
	elseif (eh <= 1.95e-83)
		tmp = abs(Float64(Float64(cos(atan(Float64(eh / Float64(ew * t)))) * ew) * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((cos(t) * eh) * tanh(asinh(((1.0 / ew) * (eh / sin(t)))))));
	tmp = 0.0;
	if (eh <= -1.7e-21)
		tmp = t_1;
	elseif (eh <= 1.95e-83)
		tmp = abs(((cos(atan((eh / (ew * t)))) * ew) * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(1.0 / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.7e-21], t$95$1, If[LessEqual[eh, 1.95e-83], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.7e-21 or 1.95e-83 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 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. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 79.6%

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

   if -1.7e-21 < eh < 1.95e-83

   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 67.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 67.6

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

   7. Applied rewrites 67.6%

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

Alternative 6: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ t_2 := \left|\left(\cos \tan^{-1} t\_1 \cdot ew\right) \cdot \sin t\right|\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-63}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;\left|\tanh t\_1 \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t))) (t_2 (fabs (* (* (cos (atan t_1)) ew) (sin t)))))
   (if (<= t -6.5e-63) t_2 (if (<= t 3.7e-37) (fabs (* (tanh t_1) eh)) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = fabs(((cos(atan(t_1)) * ew) * sin(t)));
	double tmp;
	if (t <= -6.5e-63) {
		tmp = t_2;
	} else if (t <= 3.7e-37) {
		tmp = fabs((tanh(t_1) * eh));
	} else {
		tmp = t_2;
	}
	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 / (ew * t)
    t_2 = abs(((cos(atan(t_1)) * ew) * sin(t)))
    if (t <= (-6.5d-63)) then
        tmp = t_2
    else if (t <= 3.7d-37) then
        tmp = abs((tanh(t_1) * eh))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = Math.abs(((Math.cos(Math.atan(t_1)) * ew) * Math.sin(t)));
	double tmp;
	if (t <= -6.5e-63) {
		tmp = t_2;
	} else if (t <= 3.7e-37) {
		tmp = Math.abs((Math.tanh(t_1) * eh));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * t)
	t_2 = math.fabs(((math.cos(math.atan(t_1)) * ew) * math.sin(t)))
	tmp = 0
	if t <= -6.5e-63:
		tmp = t_2
	elif t <= 3.7e-37:
		tmp = math.fabs((math.tanh(t_1) * eh))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	t_2 = abs(Float64(Float64(cos(atan(t_1)) * ew) * sin(t)))
	tmp = 0.0
	if (t <= -6.5e-63)
		tmp = t_2;
	elseif (t <= 3.7e-37)
		tmp = abs(Float64(tanh(t_1) * eh));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * t);
	t_2 = abs(((cos(atan(t_1)) * ew) * sin(t)));
	tmp = 0.0;
	if (t <= -6.5e-63)
		tmp = t_2;
	elseif (t <= 3.7e-37)
		tmp = abs((tanh(t_1) * eh));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -6.5e-63], t$95$2, If[LessEqual[t, 3.7e-37], N[Abs[N[(N[Tanh[t$95$1], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.4999999999999998e-63 or 3.7e-37 < t

   1. Initial program 99.7%

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

   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 51.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.3

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

   7. Applied rewrites 51.3%

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

   if -6.4999999999999998e-63 < t < 3.7e-37

   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 75.6%

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

   5. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in t around 0

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 75.6

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

   10. Applied rewrites 75.6%

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

Alternative 7: 42.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left|\tanh \left(\frac{eh}{ew \cdot \sin t}\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* (tanh (/ eh (* ew (sin t)))) eh)))

C

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

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((tanh((eh / (ew * sin(t)))) * eh))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Tanh[N[(eh / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * eh), $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 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 42.0%

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

5. Taylor expanded in eh around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sin.f64 42.0

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

7. Applied rewrites 42.0%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 42.0%

    \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot \sin t}\right) \cdot eh\right| \]
11. Add Preprocessing

Alternative 8: 40.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \end{array} \]

FPCore

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

C

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

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((tanh((eh / (ew * t))) * eh))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Tanh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * eh), $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 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 42.0%

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

5. Taylor expanded in eh around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sin.f64 42.0

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

7. Applied rewrites 42.0%

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

8. Taylor expanded in t around 0

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 40.1

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

10. Applied rewrites 40.1%

    \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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))))))))