Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.9s

Alternatives: 9

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* (tan t) ew))))
   (fabs
    (fma
     (* ew (sin t))
     (/ 1.0 (sqrt (+ (pow t_1 2.0) 1.0)))
     (* (* (cos t) eh) (sin (atan t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (tan(t) * ew);
	return fabs(fma((ew * sin(t)), (1.0 / sqrt((pow(t_1, 2.0) + 1.0))), ((cos(t) * eh) * sin(atan(t_1)))));
}

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 99.8

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/l/ N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 99.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. pow2 N/A

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

   9. lower-pow.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

    \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \frac{1}{\sqrt{{\left(\frac{eh}{\tan t \cdot ew}\right)}^{2} + 1}}, \left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right| \]
12. Add Preprocessing

Alternative 2: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 99.8

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/l/ N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 99.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

10. Final simplification 99.1%

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

Alternative 3: 73.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right)\\ t_2 := \cos t \cdot eh\\ \mathbf{if}\;ew \leq -2.8 \cdot 10^{-43}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{\frac{t\_2}{ew}}{\sin t}\right) \cdot ew\right) \cdot \sin t\right|\\ \mathbf{elif}\;ew \leq 3.95 \cdot 10^{+18}:\\ \;\;\;\;\left|\sin t\_1 \cdot t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t\_1 \cdot \left(ew \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (t_2 (* (cos t) eh)))
   (if (<= ew -2.8e-43)
     (fabs (* (* (cos (atan (/ (/ t_2 ew) (sin t)))) ew) (sin t)))
     (if (<= ew 3.95e+18)
       (fabs (* (sin t_1) t_2))
       (fabs (* (cos t_1) (* ew (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	double t_2 = cos(t) * eh;
	double tmp;
	if (ew <= -2.8e-43) {
		tmp = fabs(((cos(atan(((t_2 / ew) / sin(t)))) * ew) * sin(t)));
	} else if (ew <= 3.95e+18) {
		tmp = fabs((sin(t_1) * t_2));
	} else {
		tmp = fabs((cos(t_1) * (ew * sin(t))));
	}
	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 = atan((((eh / sin(t)) / ew) * cos(t)))
    t_2 = cos(t) * eh
    if (ew <= (-2.8d-43)) then
        tmp = abs(((cos(atan(((t_2 / ew) / sin(t)))) * ew) * sin(t)))
    else if (ew <= 3.95d+18) then
        tmp = abs((sin(t_1) * t_2))
    else
        tmp = abs((cos(t_1) * (ew * sin(t))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)));
	double t_2 = Math.cos(t) * eh;
	double tmp;
	if (ew <= -2.8e-43) {
		tmp = Math.abs(((Math.cos(Math.atan(((t_2 / ew) / Math.sin(t)))) * ew) * Math.sin(t)));
	} else if (ew <= 3.95e+18) {
		tmp = Math.abs((Math.sin(t_1) * t_2));
	} else {
		tmp = Math.abs((Math.cos(t_1) * (ew * Math.sin(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))
	t_2 = math.cos(t) * eh
	tmp = 0
	if ew <= -2.8e-43:
		tmp = math.fabs(((math.cos(math.atan(((t_2 / ew) / math.sin(t)))) * ew) * math.sin(t)))
	elif ew <= 3.95e+18:
		tmp = math.fabs((math.sin(t_1) * t_2))
	else:
		tmp = math.fabs((math.cos(t_1) * (ew * math.sin(t))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))
	t_2 = Float64(cos(t) * eh)
	tmp = 0.0
	if (ew <= -2.8e-43)
		tmp = abs(Float64(Float64(cos(atan(Float64(Float64(t_2 / ew) / sin(t)))) * ew) * sin(t)));
	elseif (ew <= 3.95e+18)
		tmp = abs(Float64(sin(t_1) * t_2));
	else
		tmp = abs(Float64(cos(t_1) * Float64(ew * sin(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	t_2 = cos(t) * eh;
	tmp = 0.0;
	if (ew <= -2.8e-43)
		tmp = abs(((cos(atan(((t_2 / ew) / sin(t)))) * ew) * sin(t)));
	elseif (ew <= 3.95e+18)
		tmp = abs((sin(t_1) * t_2));
	else
		tmp = abs((cos(t_1) * (ew * sin(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[ew, -2.8e-43], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(t$95$2 / ew), $MachinePrecision] / N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 3.95e+18], N[Abs[N[(N[Sin[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t$95$1], $MachinePrecision] * N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.7999999999999998e-43

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in ew around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-atan.f64 N/A

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

       7. associate-/r* N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-cos.f64 N/A

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

       13. lower-sin.f64 N/A

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

       14. lower-sin.f64 70.5

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

   11. Applied rewrites 70.5%

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

   if -2.7999999999999998e-43 < ew < 3.95e18

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 88.4%

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

   if 3.95e18 < ew

   1. Initial program 99.7%

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

   3. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 71.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.8 \cdot 10^{-43}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{\frac{\cos t \cdot eh}{ew}}{\sin t}\right) \cdot ew\right) \cdot \sin t\right|\\ \mathbf{elif}\;ew \leq 3.95 \cdot 10^{+18}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(ew \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right)\\ t_2 := \left|\cos t\_1 \cdot \left(ew \cdot \sin t\right)\right|\\ \mathbf{if}\;ew \leq -2.8 \cdot 10^{-43}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 3.95 \cdot 10^{+18}:\\ \;\;\;\;\left|\sin t\_1 \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (/ (/ eh (sin t)) ew) (cos t))))
        (t_2 (fabs (* (cos t_1) (* ew (sin t))))))
   (if (<= ew -2.8e-43)
     t_2
     (if (<= ew 3.95e+18) (fabs (* (sin t_1) (* (cos t) eh))) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	double t_2 = fabs((cos(t_1) * (ew * sin(t))));
	double tmp;
	if (ew <= -2.8e-43) {
		tmp = t_2;
	} else if (ew <= 3.95e+18) {
		tmp = fabs((sin(t_1) * (cos(t) * eh)));
	} 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 = atan((((eh / sin(t)) / ew) * cos(t)))
    t_2 = abs((cos(t_1) * (ew * sin(t))))
    if (ew <= (-2.8d-43)) then
        tmp = t_2
    else if (ew <= 3.95d+18) then
        tmp = abs((sin(t_1) * (cos(t) * eh)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)));
	double t_2 = Math.abs((Math.cos(t_1) * (ew * Math.sin(t))));
	double tmp;
	if (ew <= -2.8e-43) {
		tmp = t_2;
	} else if (ew <= 3.95e+18) {
		tmp = Math.abs((Math.sin(t_1) * (Math.cos(t) * eh)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))
	t_2 = math.fabs((math.cos(t_1) * (ew * math.sin(t))))
	tmp = 0
	if ew <= -2.8e-43:
		tmp = t_2
	elif ew <= 3.95e+18:
		tmp = math.fabs((math.sin(t_1) * (math.cos(t) * eh)))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))
	t_2 = abs(Float64(cos(t_1) * Float64(ew * sin(t))))
	tmp = 0.0
	if (ew <= -2.8e-43)
		tmp = t_2;
	elseif (ew <= 3.95e+18)
		tmp = abs(Float64(sin(t_1) * Float64(cos(t) * eh)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	t_2 = abs((cos(t_1) * (ew * sin(t))));
	tmp = 0.0;
	if (ew <= -2.8e-43)
		tmp = t_2;
	elseif (ew <= 3.95e+18)
		tmp = abs((sin(t_1) * (cos(t) * eh)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Cos[t$95$1], $MachinePrecision] * N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.8e-43], t$95$2, If[LessEqual[ew, 3.95e+18], N[Abs[N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.7999999999999998e-43 or 3.95e18 < ew

   1. Initial program 99.7%

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

   3. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   if -2.7999999999999998e-43 < ew < 3.95e18

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 88.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.8 \cdot 10^{-43}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(ew \cdot \sin t\right)\right|\\ \mathbf{elif}\;ew \leq 3.95 \cdot 10^{+18}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(ew \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(ew \cdot \sin t\right)\right|\\ \mathbf{if}\;ew \leq -2.8 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.95 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{1}{\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (cos (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (* ew (sin t))))))
   (if (<= ew -2.8e-43)
     t_1
     (if (<= ew 3.95e+18)
       (/
        1.0
        (/ 1.0 (fabs (* (* (cos t) eh) (sin (atan (/ eh (* (tan t) ew))))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (ew * sin(t))));
	double tmp;
	if (ew <= -2.8e-43) {
		tmp = t_1;
	} else if (ew <= 3.95e+18) {
		tmp = 1.0 / (1.0 / fabs(((cos(t) * eh) * sin(atan((eh / (tan(t) * 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((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (ew * sin(t))))
    if (ew <= (-2.8d-43)) then
        tmp = t_1
    else if (ew <= 3.95d+18) then
        tmp = 1.0d0 / (1.0d0 / abs(((cos(t) * eh) * sin(atan((eh / (tan(t) * 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.cos(Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)))) * (ew * Math.sin(t))));
	double tmp;
	if (ew <= -2.8e-43) {
		tmp = t_1;
	} else if (ew <= 3.95e+18) {
		tmp = 1.0 / (1.0 / Math.abs(((Math.cos(t) * eh) * Math.sin(Math.atan((eh / (Math.tan(t) * ew)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))) * (ew * math.sin(t))))
	tmp = 0
	if ew <= -2.8e-43:
		tmp = t_1
	elif ew <= 3.95e+18:
		tmp = 1.0 / (1.0 / math.fabs(((math.cos(t) * eh) * math.sin(math.atan((eh / (math.tan(t) * ew)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))) * Float64(ew * sin(t))))
	tmp = 0.0
	if (ew <= -2.8e-43)
		tmp = t_1;
	elseif (ew <= 3.95e+18)
		tmp = Float64(1.0 / Float64(1.0 / abs(Float64(Float64(cos(t) * eh) * sin(atan(Float64(eh / Float64(tan(t) * ew))))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (ew * sin(t))));
	tmp = 0.0;
	if (ew <= -2.8e-43)
		tmp = t_1;
	elseif (ew <= 3.95e+18)
		tmp = 1.0 / (1.0 / abs(((cos(t) * eh) * sin(atan((eh / (tan(t) * ew)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.8e-43], t$95$1, If[LessEqual[ew, 3.95e+18], N[(1.0 / N[(1.0 / N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.7999999999999998e-43 or 3.95e18 < ew

   1. Initial program 99.7%

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

   3. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   if -2.7999999999999998e-43 < ew < 3.95e18

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 88.3

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

   7. Applied rewrites 88.3%

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

   9. Applied rewrites 88.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.8 \cdot 10^{-43}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(ew \cdot \sin t\right)\right|\\ \mathbf{elif}\;ew \leq 3.95 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{1}{\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|}}\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(ew \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (/ 1.0 (/ 1.0 (fabs (* (* (cos t) eh) (sin (atan (/ eh (* (tan t) ew)))))))))

C

double code(double eh, double ew, double t) {
	return 1.0 / (1.0 / fabs(((cos(t) * eh) * sin(atan((eh / (tan(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 = 1.0d0 / (1.0d0 / abs(((cos(t) * eh) * sin(atan((eh / (tan(t) * ew)))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return 1.0 / (1.0 / Math.abs(((Math.cos(t) * eh) * Math.sin(Math.atan((eh / (Math.tan(t) * ew)))))));
}

Python

def code(eh, ew, t):
	return 1.0 / (1.0 / math.fabs(((math.cos(t) * eh) * math.sin(math.atan((eh / (math.tan(t) * ew)))))))

Julia

function code(eh, ew, t)
	return Float64(1.0 / Float64(1.0 / abs(Float64(Float64(cos(t) * eh) * sin(atan(Float64(eh / Float64(tan(t) * ew))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = 1.0 / (1.0 / abs(((cos(t) * eh) * sin(atan((eh / (tan(t) * ew)))))));
end

Wolfram

code[eh_, ew_, t_] := N[(1.0 / N[(1.0 / N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in ew around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-atan.f64 N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 60.6

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

7. Applied rewrites 60.6%

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

9. Applied rewrites 60.6%

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

Alternative 7: 55.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right)}{t}\right) \cdot t\_1\right|\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+137}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 4.05 \cdot 10^{+96}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2
         (fabs
          (*
           (sin
            (atan
             (/ (fma (* (* -0.3333333333333333 (/ eh ew)) t) t (/ eh ew)) t)))
           t_1))))
   (if (<= t -7.5e+137)
     t_2
     (if (<= t 4.05e+96) (fabs (* (sin (atan (/ eh (* ew t)))) t_1)) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs((sin(atan((fma(((-0.3333333333333333 * (eh / ew)) * t), t, (eh / ew)) / t))) * t_1));
	double tmp;
	if (t <= -7.5e+137) {
		tmp = t_2;
	} else if (t <= 4.05e+96) {
		tmp = fabs((sin(atan((eh / (ew * t)))) * t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(Float64(sin(atan(Float64(fma(Float64(Float64(-0.3333333333333333 * Float64(eh / ew)) * t), t, Float64(eh / ew)) / t))) * t_1))
	tmp = 0.0
	if (t <= -7.5e+137)
		tmp = t_2;
	elseif (t <= 4.05e+96)
		tmp = abs(Float64(sin(atan(Float64(eh / Float64(ew * t)))) * t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(-0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7.5e+137], t$95$2, If[LessEqual[t, 4.05e+96], N[Abs[N[(N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.50000000000000025e137 or 4.0500000000000001e96 < t

   1. Initial program 99.5%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 47.4%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\left(\frac{eh}{ew} \cdot -0.3333333333333333\right) \cdot t, t, \frac{eh}{ew}\right)}{t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]

   if -7.50000000000000025e137 < t < 4.0500000000000001e96

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 63.1%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+137}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right)}{t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{elif}\;t \leq 4.05 \cdot 10^{+96}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right)}{t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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(atan((eh / (ew * t)))) * (cos(t) * eh)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-atan.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

5. Applied rewrites 60.6%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 50.4%

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

Alternative 9: 41.6% accurate, 174.0× speedup

Math

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

FPCore

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

C

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

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(-eh)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 42.4

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

5. Applied rewrites 42.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 40.6%

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

10. Applied rewrites 11.8%

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

11. Taylor expanded in eh around -inf

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

13. Applied rewrites 42.8%

    \[\leadsto \left|-eh\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024267 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))