Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in eh around 0

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

   1. fp-cancel-sign-sub-inv N/A

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

   2. distribute-lft-neg-out N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. mul-1-neg N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. mul-1-neg N/A

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

Alternative 2: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_3 := t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2\\ t_4 := \frac{\tan t}{ew}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-87}:\\ \;\;\;\;\left(-ew\right) \cdot \cos t\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(t\_4 \cdot \left(-eh\right), \left(-eh\right) \cdot t, ew\right)}{\cosh \sinh^{-1} \left(t\_4 \cdot eh\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (atan (/ (* eh (tan t)) (- ew))))
        (t_3 (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))))
        (t_4 (/ (tan t) ew)))
   (if (<= t_3 -5e-87)
     (* (- ew) (cos t))
     (if (<= t_3 4e-43)
       (fabs
        (/ (fma (* t_4 (- eh)) (* (- eh) t) ew) (cosh (asinh (* t_4 eh)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double t_3 = (t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2));
	double t_4 = tan(t) / ew;
	double tmp;
	if (t_3 <= -5e-87) {
		tmp = -ew * cos(t);
	} else if (t_3 <= 4e-43) {
		tmp = fabs((fma((t_4 * -eh), (-eh * t), ew) / cosh(asinh((t_4 * eh)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_3 = Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2)))
	t_4 = Float64(tan(t) / ew)
	tmp = 0.0
	if (t_3 <= -5e-87)
		tmp = Float64(Float64(-ew) * cos(t));
	elseif (t_3 <= 4e-43)
		tmp = abs(Float64(fma(Float64(t_4 * Float64(-eh)), Float64(Float64(-eh) * t), ew) / cosh(asinh(Float64(t_4 * eh)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-87], N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e-43], N[Abs[N[(N[(N[(t$95$4 * (-eh)), $MachinePrecision] * N[((-eh) * t), $MachinePrecision] + ew), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$4 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -5.00000000000000042e-87

   1. Initial program 99.8%

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(ew \cdot \cos t\right) \cdot {\left(\sqrt{-1}\right)}^{2}} \]

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

         \[\leadsto \left(ew \cdot \cos t\right) \cdot \color{blue}{-1} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(ew \cdot \cos t\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot ew\right) \cdot \cos t} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(ew\right)\right)} \cdot \cos t \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(ew\right)\right) \cdot \cos t} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-ew\right)} \cdot \cos t \]

      9. lower-cos.f64 58.4

         \[\leadsto \left(-ew\right) \cdot \color{blue}{\cos t} \]

   7. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(-ew\right) \cdot \cos t} \]

   if -5.00000000000000042e-87 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 4.00000000000000031e-43

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 63.0%

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

   if 4.00000000000000031e-43 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.7%

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 32.5

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 68.3

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

   10. Applied rewrites 68.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-87}:\\ \;\;\;\;\left(-ew\right) \cdot \cos t\\ \mathbf{elif}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq 4 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\tan t}{ew} \cdot \left(-eh\right), \left(-eh\right) \cdot t, ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left(-ew\right) \cdot \cos t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -1e-283)
     (* (- ew) (cos t))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-283) {
		tmp = -ew * cos(t);
	} else {
		tmp = t_1;
	}
	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 = ew * cos(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-1d-283)) then
        tmp = -ew * cos(t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -1e-283) {
		tmp = -ew * Math.cos(t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -1e-283:
		tmp = -ew * math.cos(t)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -1e-283)
		tmp = Float64(Float64(-ew) * cos(t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-283)
		tmp = -ew * cos(t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-283], N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999947e-284

   1. Initial program 99.8%

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(ew \cdot \cos t\right) \cdot {\left(\sqrt{-1}\right)}^{2}} \]

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

         \[\leadsto \left(ew \cdot \cos t\right) \cdot \color{blue}{-1} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(ew \cdot \cos t\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot ew\right) \cdot \cos t} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(ew\right)\right)} \cdot \cos t \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(ew\right)\right) \cdot \cos t} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-ew\right)} \cdot \cos t \]

      9. lower-cos.f64 57.6

         \[\leadsto \left(-ew\right) \cdot \color{blue}{\cos t} \]

   7. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\left(-ew\right) \cdot \cos t} \]

   if -9.99999999999999947e-284 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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. Add Preprocessing

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.1

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

   7. Applied rewrites 34.1%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 65.3

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

   10. Applied rewrites 65.3%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left(-ew\right) \cdot \cos t\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. associate-/l* N/A

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

   6. distribute-lft-neg-in N/A

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

   7. associate-/l* N/A

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

   8. associate-*l/ N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lower-*.f64 N/A

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

   11. distribute-neg-frac N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in eh around 0

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

   1. fp-cancel-sign-sub-inv N/A

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

   2. distribute-lft-neg-out N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. mul-1-neg N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. mul-1-neg N/A

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 99.1%

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

12. Applied rewrites 99.1%

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

13. Final simplification 99.1%

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

Alternative 6: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in eh around 0

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

   1. fp-cancel-sign-sub-inv N/A

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

   2. distribute-lft-neg-out N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. mul-1-neg N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. mul-1-neg N/A

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

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

Alternative 7: 75.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ \mathbf{if}\;ew \leq -1.05 \cdot 10^{-119} \lor \neg \left(ew \leq 1.55 \cdot 10^{-86}\right):\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{t\_1}\right) \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew)))
   (if (or (<= ew -1.05e-119) (not (<= ew 1.55e-86)))
     (fabs (* (cos (atan (* (/ (- (sin t)) ew) (/ eh (cos t))))) t_1))
     (fabs (* (- eh) (* (sin (atan (/ (* (- eh) (sin t)) t_1))) (sin t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double tmp;
	if ((ew <= -1.05e-119) || !(ew <= 1.55e-86)) {
		tmp = fabs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * t_1));
	} else {
		tmp = fabs((-eh * (sin(atan(((-eh * sin(t)) / t_1))) * sin(t))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(t) * ew
    if ((ew <= (-1.05d-119)) .or. (.not. (ew <= 1.55d-86))) then
        tmp = abs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * t_1))
    else
        tmp = abs((-eh * (sin(atan(((-eh * sin(t)) / t_1))) * sin(t))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.cos(t) * ew;
	double tmp;
	if ((ew <= -1.05e-119) || !(ew <= 1.55e-86)) {
		tmp = Math.abs((Math.cos(Math.atan(((-Math.sin(t) / ew) * (eh / Math.cos(t))))) * t_1));
	} else {
		tmp = Math.abs((-eh * (Math.sin(Math.atan(((-eh * Math.sin(t)) / t_1))) * Math.sin(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * ew
	tmp = 0
	if (ew <= -1.05e-119) or not (ew <= 1.55e-86):
		tmp = math.fabs((math.cos(math.atan(((-math.sin(t) / ew) * (eh / math.cos(t))))) * t_1))
	else:
		tmp = math.fabs((-eh * (math.sin(math.atan(((-eh * math.sin(t)) / t_1))) * math.sin(t))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	tmp = 0.0
	if ((ew <= -1.05e-119) || !(ew <= 1.55e-86))
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(-sin(t)) / ew) * Float64(eh / cos(t))))) * t_1));
	else
		tmp = abs(Float64(Float64(-eh) * Float64(sin(atan(Float64(Float64(Float64(-eh) * sin(t)) / t_1))) * sin(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * ew;
	tmp = 0.0;
	if ((ew <= -1.05e-119) || ~((ew <= 1.55e-86)))
		tmp = abs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * t_1));
	else
		tmp = abs((-eh * (sin(atan(((-eh * sin(t)) / t_1))) * sin(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[Or[LessEqual[ew, -1.05e-119], N[Not[LessEqual[ew, 1.55e-86]], $MachinePrecision]], N[Abs[N[(N[Cos[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-eh) * N[(N[Sin[N[ArcTan[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.05e-119 or 1.54999999999999994e-86 < 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. Add Preprocessing

   3. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

   if -1.05e-119 < ew < 1.54999999999999994e-86

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. distribute-lft-neg-out N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. mul-1-neg N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 79.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.05 \cdot 10^{-119} \lor \neg \left(ew \leq 1.55 \cdot 10^{-86}\right):\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{\cos t \cdot ew}\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-18} \lor \neg \left(t \leq 14\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{\cos t \cdot ew}\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(t\_1 \cdot \left(-eh\right), \left(-eh\right) \cdot t, ew\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)))
   (if (or (<= t -2.8e-18) (not (<= t 14.0)))
     (fabs
      (*
       (- eh)
       (* (sin (atan (/ (* (- eh) (sin t)) (* (cos t) ew)))) (sin t))))
     (fabs
      (/ (fma (* t_1 (- eh)) (* (- eh) t) ew) (cosh (asinh (* t_1 eh))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double tmp;
	if ((t <= -2.8e-18) || !(t <= 14.0)) {
		tmp = fabs((-eh * (sin(atan(((-eh * sin(t)) / (cos(t) * ew)))) * sin(t))));
	} else {
		tmp = fabs((fma((t_1 * -eh), (-eh * t), ew) / cosh(asinh((t_1 * eh)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	tmp = 0.0
	if ((t <= -2.8e-18) || !(t <= 14.0))
		tmp = abs(Float64(Float64(-eh) * Float64(sin(atan(Float64(Float64(Float64(-eh) * sin(t)) / Float64(cos(t) * ew)))) * sin(t))));
	else
		tmp = abs(Float64(fma(Float64(t_1 * Float64(-eh)), Float64(Float64(-eh) * t), ew) / cosh(asinh(Float64(t_1 * eh)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[t, -2.8e-18], N[Not[LessEqual[t, 14.0]], $MachinePrecision]], N[Abs[N[((-eh) * N[(N[Sin[N[ArcTan[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$1 * (-eh)), $MachinePrecision] * N[((-eh) * t), $MachinePrecision] + ew), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.80000000000000012e-18 or 14 < t

   1. Initial program 99.6%

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

   3. Taylor expanded in eh around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. distribute-lft-neg-out N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. mul-1-neg N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 53.0%

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

   if -2.80000000000000012e-18 < t < 14

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 90.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-18} \lor \neg \left(t \leq 14\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{\cos t \cdot ew}\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\tan t}{ew} \cdot \left(-eh\right), \left(-eh\right) \cdot t, ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.75 \cdot 10^{-174}:\\ \;\;\;\;-ew\\ \mathbf{elif}\;ew \leq 4.6 \cdot 10^{-93}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.75e-174)
   (- ew)
   (if (<= ew 4.6e-93) (* eh (sin t)) (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.75e-174) {
		tmp = -ew;
	} else if (ew <= 4.6e-93) {
		tmp = eh * sin(t);
	} else {
		tmp = ew * cos(t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-1.75d-174)) then
        tmp = -ew
    else if (ew <= 4.6d-93) then
        tmp = eh * sin(t)
    else
        tmp = ew * cos(t)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.75e-174) {
		tmp = -ew;
	} else if (ew <= 4.6e-93) {
		tmp = eh * Math.sin(t);
	} else {
		tmp = ew * Math.cos(t);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.75e-174:
		tmp = -ew
	elif ew <= 4.6e-93:
		tmp = eh * math.sin(t)
	else:
		tmp = ew * math.cos(t)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.75e-174)
		tmp = Float64(-ew);
	elseif (ew <= 4.6e-93)
		tmp = Float64(eh * sin(t));
	else
		tmp = Float64(ew * cos(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.75e-174)
		tmp = -ew;
	elseif (ew <= 4.6e-93)
		tmp = eh * sin(t);
	else
		tmp = ew * cos(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.75e-174], (-ew), If[LessEqual[ew, 4.6e-93], N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.74999999999999994e-174

   1. Initial program 99.8%

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{ew \cdot {\left(\sqrt{-1}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot ew} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot ew \]

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{-1} \cdot ew \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(ew\right)} \]

      5. lower-neg.f64 55.3

         \[\leadsto \color{blue}{-ew} \]

   7. Applied rewrites 55.3%

      \[\leadsto \color{blue}{-ew} \]

   if -1.74999999999999994e-174 < ew < 4.5999999999999996e-93

   1. Initial program 99.7%

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 16.9%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 36.7%

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

   if 4.5999999999999996e-93 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 39.5

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

   7. Applied rewrites 39.5%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 60.9

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

   10. Applied rewrites 60.9%

       \[\leadsto \color{blue}{ew \cdot \cos t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 43.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.75 \cdot 10^{-174}:\\ \;\;\;\;-ew\\ \mathbf{elif}\;ew \leq 3.5 \cdot 10^{-90}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \frac{eh \cdot eh}{ew}, t \cdot t, ew\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.75e-174)
   (- ew)
   (if (<= ew 3.5e-90)
     (* eh (sin t))
     (fma (* 0.5 (/ (* eh eh) ew)) (* t t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.75e-174) {
		tmp = -ew;
	} else if (ew <= 3.5e-90) {
		tmp = eh * sin(t);
	} else {
		tmp = fma((0.5 * ((eh * eh) / ew)), (t * t), ew);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.75e-174)
		tmp = Float64(-ew);
	elseif (ew <= 3.5e-90)
		tmp = Float64(eh * sin(t));
	else
		tmp = fma(Float64(0.5 * Float64(Float64(eh * eh) / ew)), Float64(t * t), ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.75e-174], (-ew), If[LessEqual[ew, 3.5e-90], N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.74999999999999994e-174

   1. Initial program 99.8%

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{ew \cdot {\left(\sqrt{-1}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot ew} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot ew \]

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{-1} \cdot ew \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(ew\right)} \]

      5. lower-neg.f64 55.3

         \[\leadsto \color{blue}{-ew} \]

   7. Applied rewrites 55.3%

      \[\leadsto \color{blue}{-ew} \]

   if -1.74999999999999994e-174 < ew < 3.4999999999999999e-90

   1. Initial program 99.7%

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 16.9%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 36.7%

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

   if 3.4999999999999999e-90 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 39.5

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

   7. Applied rewrites 39.5%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 44.6%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot \frac{eh \cdot eh}{ew}, \color{blue}{t} \cdot t, ew\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 38.5% accurate, 22.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 2.35 \cdot 10^{-296}:\\ \;\;\;\;-ew\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \frac{eh \cdot eh}{ew}, t \cdot t, ew\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 2.35e-296) (- ew) (fma (* 0.5 (/ (* eh eh) ew)) (* t t) ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 2.35e-296) {
		tmp = -ew;
	} else {
		tmp = fma((0.5 * ((eh * eh) / ew)), (t * t), ew);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 2.35e-296)
		tmp = Float64(-ew);
	else
		tmp = fma(Float64(0.5 * Float64(Float64(eh * eh) / ew)), Float64(t * t), ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 2.35e-296], (-ew), N[(N[(0.5 * N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 2.35e-296

   1. Initial program 99.8%

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{ew \cdot {\left(\sqrt{-1}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot ew} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot ew \]

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{-1} \cdot ew \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(ew\right)} \]

      5. lower-neg.f64 41.9

         \[\leadsto \color{blue}{-ew} \]

   7. Applied rewrites 41.9%

      \[\leadsto \color{blue}{-ew} \]

   if 2.35e-296 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 32.0

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

   7. Applied rewrites 32.0%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 35.9%

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

Alternative 12: 38.6% accurate, 37.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 5.2 \cdot 10^{-237}:\\ \;\;\;\;-ew\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 5.2e-237) (- ew) (fma (* -0.5 ew) (* t t) ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 5.2e-237) {
		tmp = -ew;
	} else {
		tmp = fma((-0.5 * ew), (t * t), ew);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 5.2e-237)
		tmp = Float64(-ew);
	else
		tmp = fma(Float64(-0.5 * ew), Float64(t * t), ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 5.2e-237], (-ew), N[(N[(-0.5 * ew), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 5.2000000000000005e-237

   1. Initial program 99.8%

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{ew \cdot {\left(\sqrt{-1}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot ew} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot ew \]

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{-1} \cdot ew \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(ew\right)} \]

      5. lower-neg.f64 39.2

         \[\leadsto \color{blue}{-ew} \]

   7. Applied rewrites 39.2%

      \[\leadsto \color{blue}{-ew} \]

   if 5.2000000000000005e-237 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.5

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

   7. Applied rewrites 34.5%

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

   8. Taylor expanded in eh around 0

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

   10. Applied rewrites 36.2%

       \[\leadsto \mathsf{fma}\left(-0.5 \cdot ew, \color{blue}{t} \cdot t, ew\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 21.7% accurate, 287.3× speedup

Math

\[\begin{array}{l} \\ -ew \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return -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 = -ew
end function

Java

public static double code(double eh, double ew, double t) {
	return -ew;
}

Python

def code(eh, ew, t):
	return -ew

Julia

function code(eh, ew, t)
	return Float64(-ew)
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := (-ew)

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 36.8%

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

5. Taylor expanded in t around 0

   \[\leadsto \color{blue}{ew \cdot {\left(\sqrt{-1}\right)}^{2}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot ew} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot ew \]

   3. rem-square-sqrt N/A

      \[\leadsto \color{blue}{-1} \cdot ew \]

   4. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(ew\right)} \]

   5. lower-neg.f64 22.4

      \[\leadsto \color{blue}{-ew} \]

7. Applied rewrites 22.4%

   \[\leadsto \color{blue}{-ew} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(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)))))))