Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.9s

Alternatives: 8

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

real(8) function code(eh, ew, t)
    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

real(8) function code(eh, ew, t)
    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

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+143}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 1.12 \cdot 10^{+132}:\\ \;\;\;\;\left|\frac{t\_1 \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)) (t_2 (fabs t_1)))
   (if (<= eh -9e+143)
     t_2
     (if (<= eh 1.12e+132)
       (fabs
        (/
         (+ (* t_1 (* eh (/ (tan t) ew))) (* (cos t) ew))
         (/ 1.0 (pow (+ 1.0 (pow (/ ew (* eh (tan t))) -2.0)) -0.5))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = fabs(t_1);
	double tmp;
	if (eh <= -9e+143) {
		tmp = t_2;
	} else if (eh <= 1.12e+132) {
		tmp = fabs((((t_1 * (eh * (tan(t) / ew))) + (cos(t) * ew)) / (1.0 / pow((1.0 + pow((ew / (eh * tan(t))), -2.0)), -0.5))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    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 = sin(t) * eh
    t_2 = abs(t_1)
    if (eh <= (-9d+143)) then
        tmp = t_2
    else if (eh <= 1.12d+132) then
        tmp = abs((((t_1 * (eh * (tan(t) / ew))) + (cos(t) * ew)) / (1.0d0 / ((1.0d0 + ((ew / (eh * tan(t))) ** (-2.0d0))) ** (-0.5d0)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double t_2 = Math.abs(t_1);
	double tmp;
	if (eh <= -9e+143) {
		tmp = t_2;
	} else if (eh <= 1.12e+132) {
		tmp = Math.abs((((t_1 * (eh * (Math.tan(t) / ew))) + (Math.cos(t) * ew)) / (1.0 / Math.pow((1.0 + Math.pow((ew / (eh * Math.tan(t))), -2.0)), -0.5))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	t_2 = math.fabs(t_1)
	tmp = 0
	if eh <= -9e+143:
		tmp = t_2
	elif eh <= 1.12e+132:
		tmp = math.fabs((((t_1 * (eh * (math.tan(t) / ew))) + (math.cos(t) * ew)) / (1.0 / math.pow((1.0 + math.pow((ew / (eh * math.tan(t))), -2.0)), -0.5))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = abs(t_1)
	tmp = 0.0
	if (eh <= -9e+143)
		tmp = t_2;
	elseif (eh <= 1.12e+132)
		tmp = abs(Float64(Float64(Float64(t_1 * Float64(eh * Float64(tan(t) / ew))) + Float64(cos(t) * ew)) / Float64(1.0 / (Float64(1.0 + (Float64(ew / Float64(eh * tan(t))) ^ -2.0)) ^ -0.5))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	t_2 = abs(t_1);
	tmp = 0.0;
	if (eh <= -9e+143)
		tmp = t_2;
	elseif (eh <= 1.12e+132)
		tmp = abs((((t_1 * (eh * (tan(t) / ew))) + (cos(t) * ew)) / (1.0 / ((1.0 + ((ew / (eh * tan(t))) ^ -2.0)) ^ -0.5))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[eh, -9e+143], t$95$2, If[LessEqual[eh, 1.12e+132], N[Abs[N[(N[(N[(t$95$1 * N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Power[N[(1.0 + N[Power[N[(ew / N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999993e143 or 1.12e132 < eh

   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 30.1%

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

   6. Applied rewrites 41.5%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in eh around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 77.1

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

   9. Applied rewrites 77.1%

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

   if -8.9999999999999993e143 < eh < 1.12e132

   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 76.2%

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

   6. Applied rewrites 89.9%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+143}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 1.12 \cdot 10^{+132}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ t_2 := \cos \tan^{-1} t\_1\\ t_3 := \sin t \cdot eh\\ \mathbf{if}\;ew \leq -3.4 \cdot 10^{-66}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(-\sin t\right) \cdot \left(t\_1 \cdot eh\right)\right) \cdot t\_2\right|\\ \mathbf{elif}\;ew \leq 1.6 \cdot 10^{-76}:\\ \;\;\;\;\left|t\_3\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t\_3}{ew}, eh \cdot \tan t, \cos t \cdot ew\right)\right| \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew)))
        (t_2 (cos (atan t_1)))
        (t_3 (* (sin t) eh)))
   (if (<= ew -3.4e-66)
     (fabs (* (fma (- ew) (cos t) (* (- (sin t)) (* t_1 eh))) t_2))
     (if (<= ew 1.6e-76)
       (fabs t_3)
       (* (fabs (fma (/ t_3 ew) (* eh (tan t)) (* (cos t) ew))) t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double t_2 = cos(atan(t_1));
	double t_3 = sin(t) * eh;
	double tmp;
	if (ew <= -3.4e-66) {
		tmp = fabs((fma(-ew, cos(t), (-sin(t) * (t_1 * eh))) * t_2));
	} else if (ew <= 1.6e-76) {
		tmp = fabs(t_3);
	} else {
		tmp = fabs(fma((t_3 / ew), (eh * tan(t)), (cos(t) * ew))) * t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	t_2 = cos(atan(t_1))
	t_3 = Float64(sin(t) * eh)
	tmp = 0.0
	if (ew <= -3.4e-66)
		tmp = abs(Float64(fma(Float64(-ew), cos(t), Float64(Float64(-sin(t)) * Float64(t_1 * eh))) * t_2));
	elseif (ew <= 1.6e-76)
		tmp = abs(t_3);
	else
		tmp = Float64(abs(fma(Float64(t_3 / ew), Float64(eh * tan(t)), Float64(cos(t) * ew))) * t_2);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[ew, -3.4e-66], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision] + N[((-N[Sin[t], $MachinePrecision]) * N[(t$95$1 * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.6e-76], N[Abs[t$95$3], $MachinePrecision], N[(N[Abs[N[(N[(t$95$3 / ew), $MachinePrecision] * N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3.39999999999999997e-66

   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 99.8%

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

   6. Applied rewrites 84.6%

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

   if -3.39999999999999997e-66 < ew < 1.5999999999999999e-76

   1. Initial program 99.9%

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

   4. Applied rewrites 30.0%

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

   6. Applied rewrites 53.5%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in eh around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 76.0

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

   9. Applied rewrites 76.0%

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

   if 1.5999999999999999e-76 < 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 99.8%

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

   6. Applied rewrites 78.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.4 \cdot 10^{-66}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(-\sin t\right) \cdot \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right)\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.6 \cdot 10^{-76}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t \cdot eh}{ew}, eh \cdot \tan t, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|\mathsf{fma}\left(\frac{t\_1}{ew}, eh \cdot \tan t, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{if}\;ew \leq -3.4 \cdot 10^{-66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 1.6 \cdot 10^{-76}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2
         (*
          (fabs (fma (/ t_1 ew) (* eh (tan t)) (* (cos t) ew)))
          (cos (atan (* eh (/ (tan t) ew)))))))
   (if (<= ew -3.4e-66) t_2 (if (<= ew 1.6e-76) (fabs t_1) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = fabs(fma((t_1 / ew), (eh * tan(t)), (cos(t) * ew))) * cos(atan((eh * (tan(t) / ew))));
	double tmp;
	if (ew <= -3.4e-66) {
		tmp = t_2;
	} else if (ew <= 1.6e-76) {
		tmp = fabs(t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(abs(fma(Float64(t_1 / ew), Float64(eh * tan(t)), Float64(cos(t) * ew))) * cos(atan(Float64(eh * Float64(tan(t) / ew)))))
	tmp = 0.0
	if (ew <= -3.4e-66)
		tmp = t_2;
	elseif (ew <= 1.6e-76)
		tmp = abs(t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -3.4e-66], t$95$2, If[LessEqual[ew, 1.6e-76], N[Abs[t$95$1], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.39999999999999997e-66 or 1.5999999999999999e-76 < 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 99.8%

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

   6. Applied rewrites 81.3%

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

   if -3.39999999999999997e-66 < ew < 1.5999999999999999e-76

   1. Initial program 99.9%

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

   4. Applied rewrites 30.0%

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

   6. Applied rewrites 53.5%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in eh around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 76.0

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

   9. Applied rewrites 76.0%

      \[\leadsto \left|\color{blue}{\sin t \cdot eh}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.4 \cdot 10^{-66}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t \cdot eh}{ew}, eh \cdot \tan t, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{elif}\;ew \leq 1.6 \cdot 10^{-76}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t \cdot eh}{ew}, eh \cdot \tan t, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot eh\right|\\ \mathbf{if}\;eh \leq -2.55 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.36 \cdot 10^{-58}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(-0.5 \cdot eh\right) \cdot \left(\frac{\sin t}{ew} \cdot \tan t\right), eh, \left(-ew\right) \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) eh))))
   (if (<= eh -2.55e+79)
     t_1
     (if (<= eh 1.36e-58)
       (fabs
        (fma (* (* -0.5 eh) (* (/ (sin t) ew) (tan t))) eh (* (- ew) (cos t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * eh));
	double tmp;
	if (eh <= -2.55e+79) {
		tmp = t_1;
	} else if (eh <= 1.36e-58) {
		tmp = fabs(fma(((-0.5 * eh) * ((sin(t) / ew) * tan(t))), eh, (-ew * cos(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * eh))
	tmp = 0.0
	if (eh <= -2.55e+79)
		tmp = t_1;
	elseif (eh <= 1.36e-58)
		tmp = abs(fma(Float64(Float64(-0.5 * eh) * Float64(Float64(sin(t) / ew) * tan(t))), eh, Float64(Float64(-ew) * cos(t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.55e+79], t$95$1, If[LessEqual[eh, 1.36e-58], N[Abs[N[(N[(N[(-0.5 * eh), $MachinePrecision] * N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh + N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.5500000000000001e79 or 1.36000000000000007e-58 < eh

   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 37.9%

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

   6. Applied rewrites 57.3%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in eh around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 69.4

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

   9. Applied rewrites 69.4%

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

   if -2.5500000000000001e79 < eh < 1.36000000000000007e-58

   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 88.7%

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

   6. Applied rewrites 95.0%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 88.1%

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

   11. Applied rewrites 88.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.55 \cdot 10^{+79}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 1.36 \cdot 10^{-58}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(-0.5 \cdot eh\right) \cdot \left(\frac{\sin t}{ew} \cdot \tan t\right), eh, \left(-ew\right) \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot eh\right|\\ \mathbf{if}\;eh \leq -2.55 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.36 \cdot 10^{-58}:\\ \;\;\;\;\left|\left(-ew\right) \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) eh))))
   (if (<= eh -2.55e+79)
     t_1
     (if (<= eh 1.36e-58) (fabs (* (- ew) (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * eh));
	double tmp;
	if (eh <= -2.55e+79) {
		tmp = t_1;
	} else if (eh <= 1.36e-58) {
		tmp = fabs((-ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * eh))
    if (eh <= (-2.55d+79)) then
        tmp = t_1
    else if (eh <= 1.36d-58) then
        tmp = abs((-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 = Math.abs((Math.sin(t) * eh));
	double tmp;
	if (eh <= -2.55e+79) {
		tmp = t_1;
	} else if (eh <= 1.36e-58) {
		tmp = Math.abs((-ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * eh))
	tmp = 0
	if eh <= -2.55e+79:
		tmp = t_1
	elif eh <= 1.36e-58:
		tmp = math.fabs((-ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * eh))
	tmp = 0.0
	if (eh <= -2.55e+79)
		tmp = t_1;
	elseif (eh <= 1.36e-58)
		tmp = abs(Float64(Float64(-ew) * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * eh));
	tmp = 0.0;
	if (eh <= -2.55e+79)
		tmp = t_1;
	elseif (eh <= 1.36e-58)
		tmp = abs((-ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.55e+79], t$95$1, If[LessEqual[eh, 1.36e-58], N[Abs[N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.5500000000000001e79 or 1.36000000000000007e-58 < eh

   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 37.9%

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

   6. Applied rewrites 57.3%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in eh around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 69.4

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

   9. Applied rewrites 69.4%

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

   if -2.5500000000000001e79 < eh < 1.36000000000000007e-58

   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 88.7%

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

   6. Applied rewrites 95.0%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 87.9

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

   9. Applied rewrites 87.9%

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

Alternative 7: 57.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot eh\right|\\ \mathbf{if}\;eh \leq -2.5 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.35 \cdot 10^{-58}:\\ \;\;\;\;\left|-ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) eh))))
   (if (<= eh -2.5e+79) t_1 (if (<= eh 1.35e-58) (fabs (- ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * eh));
	double tmp;
	if (eh <= -2.5e+79) {
		tmp = t_1;
	} else if (eh <= 1.35e-58) {
		tmp = fabs(-ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * eh))
    if (eh <= (-2.5d+79)) then
        tmp = t_1
    else if (eh <= 1.35d-58) then
        tmp = abs(-ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * eh));
	double tmp;
	if (eh <= -2.5e+79) {
		tmp = t_1;
	} else if (eh <= 1.35e-58) {
		tmp = Math.abs(-ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * eh))
	tmp = 0
	if eh <= -2.5e+79:
		tmp = t_1
	elif eh <= 1.35e-58:
		tmp = math.fabs(-ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * eh))
	tmp = 0.0
	if (eh <= -2.5e+79)
		tmp = t_1;
	elseif (eh <= 1.35e-58)
		tmp = abs(Float64(-ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * eh));
	tmp = 0.0;
	if (eh <= -2.5e+79)
		tmp = t_1;
	elseif (eh <= 1.35e-58)
		tmp = abs(-ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.5e+79], t$95$1, If[LessEqual[eh, 1.35e-58], N[Abs[(-ew)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.5e79 or 1.3499999999999999e-58 < eh

   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 37.9%

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

   6. Applied rewrites 57.3%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in eh around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 69.4

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

   9. Applied rewrites 69.4%

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

   if -2.5e79 < eh < 1.3499999999999999e-58

   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 88.7%

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

   6. Applied rewrites 95.0%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{-1 \cdot ew}\right| \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 57.3

         \[\leadsto \left|\color{blue}{-ew}\right| \]

   9. Applied rewrites 57.3%

      \[\leadsto \left|\color{blue}{-ew}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 41.9% accurate, 172.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 62.5%

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

6. Applied rewrites 75.6%

   \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]

7. Taylor expanded in t around 0

   \[\leadsto \left|\color{blue}{-1 \cdot ew}\right| \]
8. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 41.2

      \[\leadsto \left|\color{blue}{-ew}\right| \]

9. Applied rewrites 41.2%

   \[\leadsto \left|\color{blue}{-ew}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024264 
(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)))))))