Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.3s

Alternatives: 12

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

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* (tan t) ew))))
   (fabs
    (fma
     (/ ew (sqrt (+ 1.0 (pow t_1 2.0))))
     (sin t)
     (* (* (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 / sqrt((1.0 + pow(t_1, 2.0)))), sin(t), ((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 / sqrt(Float64(1.0 + (t_1 ^ 2.0)))), sin(t), 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[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[t], $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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 99.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 3: 90.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\\ t_3 := \left|\sin t\_2 \cdot t\_1 + \left(\sin t \cdot ew\right) \cdot \cos t\_2\right|\\ \mathbf{if}\;ew \leq -1.12 \cdot 10^{-125}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;ew \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2 (atan (/ eh (* t ew))))
        (t_3 (fabs (+ (* (sin t_2) t_1) (* (* (sin t) ew) (cos t_2))))))
   (if (<= ew -1.12e-125)
     t_3
     (if (<= ew 7.5e-134)
       (fabs (* (sin (atan (* (/ (/ eh (sin t)) ew) (cos t)))) t_1))
       t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = atan((eh / (t * ew)));
	double t_3 = fabs(((sin(t_2) * t_1) + ((sin(t) * ew) * cos(t_2))));
	double tmp;
	if (ew <= -1.12e-125) {
		tmp = t_3;
	} else if (ew <= 7.5e-134) {
		tmp = fabs((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * t_1));
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = cos(t) * eh
    t_2 = atan((eh / (t * ew)))
    t_3 = abs(((sin(t_2) * t_1) + ((sin(t) * ew) * cos(t_2))))
    if (ew <= (-1.12d-125)) then
        tmp = t_3
    else if (ew <= 7.5d-134) then
        tmp = abs((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * t_1))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.cos(t) * eh;
	double t_2 = Math.atan((eh / (t * ew)));
	double t_3 = Math.abs(((Math.sin(t_2) * t_1) + ((Math.sin(t) * ew) * Math.cos(t_2))));
	double tmp;
	if (ew <= -1.12e-125) {
		tmp = t_3;
	} else if (ew <= 7.5e-134) {
		tmp = Math.abs((Math.sin(Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)))) * t_1));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * eh
	t_2 = math.atan((eh / (t * ew)))
	t_3 = math.fabs(((math.sin(t_2) * t_1) + ((math.sin(t) * ew) * math.cos(t_2))))
	tmp = 0
	if ew <= -1.12e-125:
		tmp = t_3
	elif ew <= 7.5e-134:
		tmp = math.fabs((math.sin(math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))) * t_1))
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = atan(Float64(eh / Float64(t * ew)))
	t_3 = abs(Float64(Float64(sin(t_2) * t_1) + Float64(Float64(sin(t) * ew) * cos(t_2))))
	tmp = 0.0
	if (ew <= -1.12e-125)
		tmp = t_3;
	elseif (ew <= 7.5e-134)
		tmp = abs(Float64(sin(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))) * t_1));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * eh;
	t_2 = atan((eh / (t * ew)));
	t_3 = abs(((sin(t_2) * t_1) + ((sin(t) * ew) * cos(t_2))));
	tmp = 0.0;
	if (ew <= -1.12e-125)
		tmp = t_3;
	elseif (ew <= 7.5e-134)
		tmp = abs((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * t_1));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(N[Sin[t$95$2], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.12e-125], t$95$3, If[LessEqual[ew, 7.5e-134], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.11999999999999997e-125 or 7.50000000000000048e-134 < ew

   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

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

   if -1.11999999999999997e-125 < ew < 7.50000000000000048e-134

   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 91.6%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.12 \cdot 10^{-125}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right) + \left(\sin t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{elif}\;ew \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right) + \left(\sin t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{if}\;eh \leq -1.9 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;\left|\left(\sin t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (sin (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (* (cos t) eh)))))
   (if (<= eh -1.9e-83)
     t_1
     (if (<= eh 1.4e-112)
       (fabs (* (* (sin t) ew) (cos (atan (/ eh (* t ew))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * (cos(t) * eh)));
	double tmp;
	if (eh <= -1.9e-83) {
		tmp = t_1;
	} else if (eh <= 1.4e-112) {
		tmp = fabs(((sin(t) * ew) * cos(atan((eh / (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((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * (cos(t) * eh)))
    if (eh <= (-1.9d-83)) then
        tmp = t_1
    else if (eh <= 1.4d-112) then
        tmp = abs(((sin(t) * ew) * cos(atan((eh / (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.sin(Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)))) * (Math.cos(t) * eh)));
	double tmp;
	if (eh <= -1.9e-83) {
		tmp = t_1;
	} else if (eh <= 1.4e-112) {
		tmp = Math.abs(((Math.sin(t) * ew) * Math.cos(Math.atan((eh / (t * ew))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))) * (math.cos(t) * eh)))
	tmp = 0
	if eh <= -1.9e-83:
		tmp = t_1
	elif eh <= 1.4e-112:
		tmp = math.fabs(((math.sin(t) * ew) * math.cos(math.atan((eh / (t * ew))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))) * Float64(cos(t) * eh)))
	tmp = 0.0
	if (eh <= -1.9e-83)
		tmp = t_1;
	elseif (eh <= 1.4e-112)
		tmp = abs(Float64(Float64(sin(t) * ew) * cos(atan(Float64(eh / Float64(t * ew))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * (cos(t) * eh)));
	tmp = 0.0;
	if (eh <= -1.9e-83)
		tmp = t_1;
	elseif (eh <= 1.4e-112)
		tmp = abs(((sin(t) * ew) * cos(atan((eh / (t * ew))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.9e-83], t$95$1, If[LessEqual[eh, 1.4e-112], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.89999999999999988e-83 or 1.40000000000000011e-112 < eh

   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 80.0%

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

   if -1.89999999999999988e-83 < eh < 1.40000000000000011e-112

   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around inf

      \[\leadsto \left|\color{blue}{ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   6. 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. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 81.7

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 81.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.9 \cdot 10^{-83}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{elif}\;eh \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;\left|\left(\sin t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, \frac{t \cdot t}{ew}, \frac{-0.022222222222222223}{ew}\right), t \cdot t, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t} \cdot eh\right) \cdot eh\right|\\ \mathbf{if}\;eh \leq -7 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 5.4 \cdot 10^{-102}:\\ \;\;\;\;\left|\left(\sin t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (sin
            (atan
             (*
              (/
               (fma
                (fma
                 (fma
                  -0.0021164021164021165
                  (/ (* t t) ew)
                  (/ -0.022222222222222223 ew))
                 (* t t)
                 (/ -0.3333333333333333 ew))
                (* t t)
                (/ 1.0 ew))
               t)
              eh)))
           eh))))
   (if (<= eh -7e-59)
     t_1
     (if (<= eh 5.4e-102)
       (fabs (* (* (sin t) ew) (cos (atan (/ eh (* t ew))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((fma(fma(fma(-0.0021164021164021165, ((t * t) / ew), (-0.022222222222222223 / ew)), (t * t), (-0.3333333333333333 / ew)), (t * t), (1.0 / ew)) / t) * eh))) * eh));
	double tmp;
	if (eh <= -7e-59) {
		tmp = t_1;
	} else if (eh <= 5.4e-102) {
		tmp = fabs(((sin(t) * ew) * cos(atan((eh / (t * ew))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(fma(fma(fma(-0.0021164021164021165, Float64(Float64(t * t) / ew), Float64(-0.022222222222222223 / ew)), Float64(t * t), Float64(-0.3333333333333333 / ew)), Float64(t * t), Float64(1.0 / ew)) / t) * eh))) * eh))
	tmp = 0.0
	if (eh <= -7e-59)
		tmp = t_1;
	elseif (eh <= 5.4e-102)
		tmp = abs(Float64(Float64(sin(t) * ew) * cos(atan(Float64(eh / Float64(t * ew))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(-0.0021164021164021165 * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] + N[(-0.022222222222222223 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(-0.3333333333333333 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(1.0 / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -7e-59], t$95$1, If[LessEqual[eh, 5.4e-102], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.0000000000000002e-59 or 5.4e-102 < eh

   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 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 51.4

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

   5. Applied rewrites 51.4%

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

   7. Applied rewrites 51.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 48.9%

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

   11. Taylor expanded in t around 0

       \[\leadsto \left|\sin \tan^{-1} \left(eh \cdot \frac{{t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{-2}{945} \cdot \frac{{t}^{2}}{ew} - \frac{1}{45} \cdot \frac{1}{ew}\right) - \frac{1}{3} \cdot \frac{1}{ew}\right) + \frac{1}{ew}}{t}\right) \cdot eh\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 51.6%

       \[\leadsto \left|\sin \tan^{-1} \left(eh \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, \frac{t \cdot t}{ew}, \frac{-0.022222222222222223}{ew}\right), t \cdot t, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t}\right) \cdot eh\right| \]

   if -7.0000000000000002e-59 < eh < 5.4e-102

   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around inf

      \[\leadsto \left|\color{blue}{ew \cdot \left(\cos \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot \sin t\right)}\right| \]
   6. 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. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 78.6

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7 \cdot 10^{-59}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, \frac{t \cdot t}{ew}, \frac{-0.022222222222222223}{ew}\right), t \cdot t, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t} \cdot eh\right) \cdot eh\right|\\ \mathbf{elif}\;eh \leq 5.4 \cdot 10^{-102}:\\ \;\;\;\;\left|\left(\sin t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, \frac{t \cdot t}{ew}, \frac{-0.022222222222222223}{ew}\right), t \cdot t, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t} \cdot eh\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, \frac{t \cdot t}{ew}, \frac{-0.022222222222222223}{ew}\right), t \cdot t, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t} \cdot eh\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (fma
        (fma
         (fma
          -0.0021164021164021165
          (/ (* t t) ew)
          (/ -0.022222222222222223 ew))
         (* t t)
         (/ -0.3333333333333333 ew))
        (* t t)
        (/ 1.0 ew))
       t)
      eh)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((fma(fma(fma(-0.0021164021164021165, ((t * t) / ew), (-0.022222222222222223 / ew)), (t * t), (-0.3333333333333333 / ew)), (t * t), (1.0 / ew)) / t) * eh))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(fma(fma(fma(-0.0021164021164021165, Float64(Float64(t * t) / ew), Float64(-0.022222222222222223 / ew)), Float64(t * t), Float64(-0.3333333333333333 / ew)), Float64(t * t), Float64(1.0 / ew)) / t) * eh))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(-0.0021164021164021165 * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] + N[(-0.022222222222222223 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(-0.3333333333333333 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(1.0 / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. 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 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.2%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 35.3%

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

11. Taylor expanded in t around 0

    \[\leadsto \left|\sin \tan^{-1} \left(eh \cdot \frac{{t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{-2}{945} \cdot \frac{{t}^{2}}{ew} - \frac{1}{45} \cdot \frac{1}{ew}\right) - \frac{1}{3} \cdot \frac{1}{ew}\right) + \frac{1}{ew}}{t}\right) \cdot eh\right| \]
12. Step-by-step derivation

13. Applied rewrites 37.4%

    \[\leadsto \left|\sin \tan^{-1} \left(eh \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, \frac{t \cdot t}{ew}, \frac{-0.022222222222222223}{ew}\right), t \cdot t, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t}\right) \cdot eh\right| \]

14. Final simplification 37.4%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, \frac{t \cdot t}{ew}, \frac{-0.022222222222222223}{ew}\right), t \cdot t, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t} \cdot eh\right) \cdot eh\right| \]
15. Add Preprocessing

Alternative 7: 42.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223 \cdot t, \frac{t}{ew}, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t} \cdot eh\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (fma
        (fma (* -0.022222222222222223 t) (/ t ew) (/ -0.3333333333333333 ew))
        (* t t)
        (/ 1.0 ew))
       t)
      eh)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((fma(fma((-0.022222222222222223 * t), (t / ew), (-0.3333333333333333 / ew)), (t * t), (1.0 / ew)) / t) * eh))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(fma(fma(Float64(-0.022222222222222223 * t), Float64(t / ew), Float64(-0.3333333333333333 / ew)), Float64(t * t), Float64(1.0 / ew)) / t) * eh))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(-0.022222222222222223 * t), $MachinePrecision] * N[(t / ew), $MachinePrecision] + N[(-0.3333333333333333 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(1.0 / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. 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 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.2%

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

8. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(eh \cdot \frac{{t}^{2} \cdot \left(\frac{-1}{45} \cdot \frac{{t}^{2}}{ew} - \frac{1}{3} \cdot \frac{1}{ew}\right) + \frac{1}{ew}}{t}\right) \cdot eh\right| \]
9. Step-by-step derivation

10. Applied rewrites 37.4%

    \[\leadsto \left|\sin \tan^{-1} \left(eh \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223 \cdot t, \frac{t}{ew}, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t}\right) \cdot eh\right| \]

11. Final simplification 37.4%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223 \cdot t, \frac{t}{ew}, \frac{-0.3333333333333333}{ew}\right), t \cdot t, \frac{1}{ew}\right)}{t} \cdot eh\right) \cdot eh\right| \]
12. Add Preprocessing

Alternative 8: 42.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\left(\left(\frac{\frac{1}{t \cdot ew}}{t} - \frac{0.3333333333333333}{ew}\right) \cdot t\right) \cdot eh\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan (* (* (- (/ (/ 1.0 (* t ew)) t) (/ 0.3333333333333333 ew)) t) eh)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((((1.0 / (t * ew)) / t) - (0.3333333333333333 / ew)) * t) * eh))) * 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((((((1.0d0 / (t * ew)) / t) - (0.3333333333333333d0 / ew)) * t) * eh))) * eh))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((Math.sin(Math.atan((((((1.0 / (t * ew)) / t) - (0.3333333333333333 / ew)) * t) * eh))) * eh));
}

Python

def code(eh, ew, t):
	return math.fabs((math.sin(math.atan((((((1.0 / (t * ew)) / t) - (0.3333333333333333 / ew)) * t) * eh))) * eh))

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(Float64(Float64(1.0 / Float64(t * ew)) / t) - Float64(0.3333333333333333 / ew)) * t) * eh))) * eh))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((sin(atan((((((1.0 / (t * ew)) / t) - (0.3333333333333333 / ew)) * t) * eh))) * eh));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(1.0 / N[(t * ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(0.3333333333333333 / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. 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 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.2%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 37.4%

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

11. Taylor expanded in t around inf

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

13. Applied rewrites 37.4%

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

14. Final simplification 37.4%

    \[\leadsto \left|\sin \tan^{-1} \left(\left(\left(\frac{\frac{1}{t \cdot ew}}{t} - \frac{0.3333333333333333}{ew}\right) \cdot t\right) \cdot eh\right) \cdot eh\right| \]
15. Add Preprocessing

Alternative 9: 42.1% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan (* (/ (fma (/ (* t t) ew) -0.3333333333333333 (/ 1.0 ew)) t) eh)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((fma(((t * t) / ew), -0.3333333333333333, (1.0 / ew)) / t) * eh))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(fma(Float64(Float64(t * t) / ew), -0.3333333333333333, Float64(1.0 / ew)) / t) * eh))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] * -0.3333333333333333 + N[(1.0 / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. 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 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.2%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 37.4%

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

11. Final simplification 37.4%

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

Alternative 10: 42.1% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin (atan (* (/ (/ (fma (* t t) -0.3333333333333333 1.0) t) ew) eh)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma((t * t), -0.3333333333333333, 1.0) / t) / ew) * eh))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(Float64(t * t), -0.3333333333333333, 1.0) / t) / ew) * eh))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. 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 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.2%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 37.4%

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

11. Taylor expanded in ew around 0

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

13. Applied rewrites 37.4%

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

14. Final simplification 37.4%

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

Alternative 11: 40.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. 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 37.2

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

5. Applied rewrites 37.2%

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

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 35.3%

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

10. Applied rewrites 35.4%

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

Alternative 12: 39.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. 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 37.2

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

5. Applied rewrites 37.2%

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

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 35.3%

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

9. Final simplification 35.3%

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

Reproduce

herbie shell --seed 2024255 
(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))))))))