Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.8s

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

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh * Float64(tan(t) / ew)))
	return abs(fma(Float64(sin(t_1) * eh), sin(t), Float64(Float64(cos(t) * ew) * cos(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[Sin[t$95$1], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

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 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 t around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 3: 84.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot eh\right|\\ \mathbf{if}\;eh \leq -7.5 \cdot 10^{+206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.6 \cdot 10^{+223}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(eh \cdot \frac{\tan t}{ew}, \left(-\sin t\right) \cdot eh, \left(-ew\right) \cdot \cos t\right)}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) eh))))
   (if (<= eh -7.5e+206)
     t_1
     (if (<= eh 6.6e+223)
       (fabs
        (/
         (fma (* eh (/ (tan t) ew)) (* (- (sin t)) eh) (* (- ew) (cos t)))
         (/ 1.0 (pow (+ 1.0 (pow (/ ew (* eh (tan t))) -2.0)) -0.5))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * eh));
	double tmp;
	if (eh <= -7.5e+206) {
		tmp = t_1;
	} else if (eh <= 6.6e+223) {
		tmp = fabs((fma((eh * (tan(t) / ew)), (-sin(t) * eh), (-ew * cos(t))) / (1.0 / pow((1.0 + pow((ew / (eh * tan(t))), -2.0)), -0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * eh))
	tmp = 0.0
	if (eh <= -7.5e+206)
		tmp = t_1;
	elseif (eh <= 6.6e+223)
		tmp = abs(Float64(fma(Float64(eh * Float64(tan(t) / ew)), Float64(Float64(-sin(t)) * eh), Float64(Float64(-ew) * cos(t))) / Float64(1.0 / (Float64(1.0 + (Float64(ew / Float64(eh * tan(t))) ^ -2.0)) ^ -0.5))));
	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, -7.5e+206], t$95$1, If[LessEqual[eh, 6.6e+223], N[Abs[N[(N[(N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision] + N[((-ew) * N[Cos[t], $MachinePrecision]), $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$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.49999999999999958e206 or 6.5999999999999999e223 < 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 16.9%

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

   6. Applied rewrites 14.0%

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

   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 90.4

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

   9. Applied rewrites 90.4%

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

   if -7.49999999999999958e206 < eh < 6.5999999999999999e223

   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 72.3%

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

   6. Applied rewrites 86.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

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

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

      10. lift-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 86.9

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

   8. Applied rewrites 86.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.5 \cdot 10^{+206}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 6.6 \cdot 10^{+223}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(eh \cdot \frac{\tan t}{ew}, \left(-\sin t\right) \cdot eh, \left(-ew\right) \cdot \cos t\right)}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;eh \leq -7.5 \cdot 10^{+206}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 6.6 \cdot 10^{+223}:\\ \;\;\;\;\left|\frac{t\_1 \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)) (t_2 (fabs t_1)))
   (if (<= eh -7.5e+206)
     t_2
     (if (<= eh 6.6e+223)
       (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 <= -7.5e+206) {
		tmp = t_2;
	} else if (eh <= 6.6e+223) {
		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 <= (-7.5d+206)) then
        tmp = t_2
    else if (eh <= 6.6d+223) 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 <= -7.5e+206) {
		tmp = t_2;
	} else if (eh <= 6.6e+223) {
		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 <= -7.5e+206:
		tmp = t_2
	elif eh <= 6.6e+223:
		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 <= -7.5e+206)
		tmp = t_2;
	elseif (eh <= 6.6e+223)
		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 <= -7.5e+206)
		tmp = t_2;
	elseif (eh <= 6.6e+223)
		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, -7.5e+206], t$95$2, If[LessEqual[eh, 6.6e+223], 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 < -7.49999999999999958e206 or 6.5999999999999999e223 < 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 16.9%

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

   6. Applied rewrites 14.0%

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

   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 90.4

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

   9. Applied rewrites 90.4%

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

   if -7.49999999999999958e206 < eh < 6.5999999999999999e223

   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 72.3%

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

   6. Applied rewrites 86.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.5 \cdot 10^{+206}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 6.6 \cdot 10^{+223}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|\mathsf{fma}\left(\frac{t\_1}{ew}, eh \cdot \tan t, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{if}\;ew \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 5.2 \cdot 10^{-191}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -1.15e-68) t_2 (if (<= ew 5.2e-191) (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 <= -1.15e-68) {
		tmp = t_2;
	} else if (ew <= 5.2e-191) {
		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 <= -1.15e-68)
		tmp = t_2;
	elseif (ew <= 5.2e-191)
		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, -1.15e-68], t$95$2, If[LessEqual[ew, 5.2e-191], N[Abs[t$95$1], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.14999999999999998e-68 or 5.19999999999999972e-191 < 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 99.7%

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

   6. Applied rewrites 77.1%

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

   if -1.14999999999999998e-68 < ew < 5.19999999999999972e-191

   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 24.1%

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

   6. Applied rewrites 46.7%

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

   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 83.8

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

   9. Applied rewrites 83.8%

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

Alternative 6: 72.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot eh\right|\\ \mathbf{if}\;eh \leq -1.1 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\left|\left(-ew\right) \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) eh))))
   (if (<= eh -1.1e+127)
     t_1
     (if (<= eh 1.25e-29) (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 <= -1.1e+127) {
		tmp = t_1;
	} else if (eh <= 1.25e-29) {
		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 <= (-1.1d+127)) then
        tmp = t_1
    else if (eh <= 1.25d-29) 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 <= -1.1e+127) {
		tmp = t_1;
	} else if (eh <= 1.25e-29) {
		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 <= -1.1e+127:
		tmp = t_1
	elif eh <= 1.25e-29:
		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 <= -1.1e+127)
		tmp = t_1;
	elseif (eh <= 1.25e-29)
		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 <= -1.1e+127)
		tmp = t_1;
	elseif (eh <= 1.25e-29)
		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, -1.1e+127], t$95$1, If[LessEqual[eh, 1.25e-29], N[Abs[N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.1000000000000001e127 or 1.24999999999999996e-29 < 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 35.0%

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

   6. Applied rewrites 49.7%

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

   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 71.6

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

   9. Applied rewrites 71.6%

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

   if -1.1000000000000001e127 < eh < 1.24999999999999996e-29

   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 83.8%

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

   6. Applied rewrites 93.8%

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

   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 82.9

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

   9. Applied rewrites 82.9%

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

Alternative 7: 61.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot eh\right|\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;\left|-ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) eh))))
   (if (<= t -2.5e-10) t_1 (if (<= t 5.8e-11) (fabs (- ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * eh));
	double tmp;
	if (t <= -2.5e-10) {
		tmp = t_1;
	} else if (t <= 5.8e-11) {
		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 (t <= (-2.5d-10)) then
        tmp = t_1
    else if (t <= 5.8d-11) 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 (t <= -2.5e-10) {
		tmp = t_1;
	} else if (t <= 5.8e-11) {
		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 t <= -2.5e-10:
		tmp = t_1
	elif t <= 5.8e-11:
		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 (t <= -2.5e-10)
		tmp = t_1;
	elseif (t <= 5.8e-11)
		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 (t <= -2.5e-10)
		tmp = t_1;
	elseif (t <= 5.8e-11)
		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[t, -2.5e-10], t$95$1, If[LessEqual[t, 5.8e-11], N[Abs[(-ew)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.50000000000000016e-10 or 5.8e-11 < 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

   4. Applied rewrites 53.0%

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

   6. Applied rewrites 65.9%

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

   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 53.4

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

   9. Applied rewrites 53.4%

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

   if -2.50000000000000016e-10 < t < 5.8e-11

   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

   4. Applied rewrites 78.3%

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

   6. Applied rewrites 88.9%

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

   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 77.9

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

   9. Applied rewrites 77.9%

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

Alternative 8: 42.1% 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 99.7%

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

4. Applied rewrites 63.6%

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

6. Applied rewrites 75.5%

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

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 40.6

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

9. Applied rewrites 40.6%

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

Reproduce

herbie shell --seed 2024276 
(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)))))))