Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

Alternatives: 10

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

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/l/ N/A

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

8. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. 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%

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

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: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. lower-*.f64 98.8

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

5. Applied rewrites 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 74.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{if}\;eh \leq -2.25 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.2 \cdot 10^{-62}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (sin (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (* (cos t) eh)))))
   (if (<= eh -2.25e-79) t_1 (if (<= eh 1.2e-62) (fabs (* (sin 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 <= -2.25e-79) {
		tmp = t_1;
	} else if (eh <= 1.2e-62) {
		tmp = fabs((sin(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 <= (-2.25d-79)) then
        tmp = t_1
    else if (eh <= 1.2d-62) then
        tmp = abs((sin(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 <= -2.25e-79) {
		tmp = t_1;
	} else if (eh <= 1.2e-62) {
		tmp = Math.abs((Math.sin(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 <= -2.25e-79:
		tmp = t_1
	elif eh <= 1.2e-62:
		tmp = math.fabs((math.sin(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 <= -2.25e-79)
		tmp = t_1;
	elseif (eh <= 1.2e-62)
		tmp = abs(Float64(sin(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 <= -2.25e-79)
		tmp = t_1;
	elseif (eh <= 1.2e-62)
		tmp = abs((sin(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, -2.25e-79], t$95$1, If[LessEqual[eh, 1.2e-62], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.2500000000000001e-79 or 1.19999999999999992e-62 < 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 eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   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 82.5%

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

   if -2.2500000000000001e-79 < eh < 1.19999999999999992e-62

   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

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 72.8

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

   11. Applied rewrites 72.8%

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

Alternative 4: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	return abs(fma(Float64(ew / 1.0), 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[(ew / 1.0), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/l/ N/A

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in eh around 0

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

11. Applied rewrites 98.7%

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

12. Final simplification 98.7%

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

Alternative 5: 61.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -0.0013:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{1}{\frac{t}{\mathsf{fma}\left(t \cdot t, -0.3333333333333333, 1\right) \cdot eh} \cdot ew}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -0.0013)
     t_1
     (if (<= t 4.5e-38)
       (fabs
        (*
         (sin
          (atan
           (/ 1.0 (* (/ t (* (fma (* t t) -0.3333333333333333 1.0) eh)) ew))))
         eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -0.0013) {
		tmp = t_1;
	} else if (t <= 4.5e-38) {
		tmp = fabs((sin(atan((1.0 / ((t / (fma((t * t), -0.3333333333333333, 1.0) * eh)) * ew)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -0.0013)
		tmp = t_1;
	elseif (t <= 4.5e-38)
		tmp = abs(Float64(sin(atan(Float64(1.0 / Float64(Float64(t / Float64(fma(Float64(t * t), -0.3333333333333333, 1.0) * eh)) * ew)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.0013], t$95$1, If[LessEqual[t, 4.5e-38], N[Abs[N[(N[Sin[N[ArcTan[N[(1.0 / N[(N[(t / N[(N[(N[(t * t), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0012999999999999999 or 4.50000000000000009e-38 < t

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 99.7%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot ew, \sin t, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \left(\cos t \cdot eh\right)\right)}\right| \]
   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 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 50.9

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

   11. Applied rewrites 50.9%

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

   if -0.0012999999999999999 < t < 4.50000000000000009e-38

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.1%

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

   10. Applied rewrites 77.4%

       \[\leadsto \left|\sin \tan^{-1} \left(\frac{1}{\frac{t}{\mathsf{fma}\left(t \cdot t, -0.3333333333333333, 1\right) \cdot eh} \cdot ew}\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 61.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -0.0013:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.3333333333333333, 1\right) \cdot eh}{t \cdot ew}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -0.0013)
     t_1
     (if (<= t 4.5e-38)
       (fabs
        (*
         (sin (atan (/ (* (fma (* t t) -0.3333333333333333 1.0) eh) (* t ew))))
         eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -0.0013) {
		tmp = t_1;
	} else if (t <= 4.5e-38) {
		tmp = fabs((sin(atan(((fma((t * t), -0.3333333333333333, 1.0) * eh) / (t * ew)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -0.0013)
		tmp = t_1;
	elseif (t <= 4.5e-38)
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(Float64(t * t), -0.3333333333333333, 1.0) * eh) / Float64(t * ew)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.0013], t$95$1, If[LessEqual[t, 4.5e-38], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(t * t), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * eh), $MachinePrecision] / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0012999999999999999 or 4.50000000000000009e-38 < t

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 99.7%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot ew, \sin t, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \left(\cos t \cdot eh\right)\right)}\right| \]
   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 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 50.9

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

   11. Applied rewrites 50.9%

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

   if -0.0012999999999999999 < t < 4.50000000000000009e-38

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.1%

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

   10. Applied rewrites 77.4%

       \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.3333333333333333, 1\right) \cdot eh}{t \cdot ew}\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -0.0013:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -0.0013)
     t_1
     (if (<= t 4.5e-38) (fabs (* (sin (atan (/ (/ eh ew) t))) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -0.0013) {
		tmp = t_1;
	} else if (t <= 4.5e-38) {
		tmp = fabs((sin(atan(((eh / ew) / t))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (t <= (-0.0013d0)) then
        tmp = t_1
    else if (t <= 4.5d-38) then
        tmp = abs((sin(atan(((eh / ew) / t))) * eh))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (t <= -0.0013) {
		tmp = t_1;
	} else if (t <= 4.5e-38) {
		tmp = Math.abs((Math.sin(Math.atan(((eh / ew) / t))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if t <= -0.0013:
		tmp = t_1
	elif t <= 4.5e-38:
		tmp = math.fabs((math.sin(math.atan(((eh / ew) / t))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -0.0013)
		tmp = t_1;
	elseif (t <= 4.5e-38)
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / ew) / t))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (t <= -0.0013)
		tmp = t_1;
	elseif (t <= 4.5e-38)
		tmp = abs((sin(atan(((eh / ew) / t))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.0013], t$95$1, If[LessEqual[t, 4.5e-38], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0012999999999999999 or 4.50000000000000009e-38 < t

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 99.7%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot ew, \sin t, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \left(\cos t \cdot eh\right)\right)}\right| \]
   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 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 50.9

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

   11. Applied rewrites 50.9%

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

   if -0.0012999999999999999 < t < 4.50000000000000009e-38

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   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 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 77.4%

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

Alternative 8: 50.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{t \cdot eh}{ew} \cdot -0.3333333333333333\right) \cdot eh\right|\\ \mathbf{if}\;eh \leq -1.42 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+132}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs (* (sin (atan (* (/ (* t eh) ew) -0.3333333333333333))) eh))))
   (if (<= eh -1.42e+116)
     t_1
     (if (<= eh 4.8e+132) (fabs (* (sin t) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan((((t * eh) / ew) * -0.3333333333333333))) * eh));
	double tmp;
	if (eh <= -1.42e+116) {
		tmp = t_1;
	} else if (eh <= 4.8e+132) {
		tmp = fabs((sin(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((((t * eh) / ew) * (-0.3333333333333333d0)))) * eh))
    if (eh <= (-1.42d+116)) then
        tmp = t_1
    else if (eh <= 4.8d+132) then
        tmp = abs((sin(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((((t * eh) / ew) * -0.3333333333333333))) * eh));
	double tmp;
	if (eh <= -1.42e+116) {
		tmp = t_1;
	} else if (eh <= 4.8e+132) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan((((t * eh) / ew) * -0.3333333333333333))) * eh))
	tmp = 0
	if eh <= -1.42e+116:
		tmp = t_1
	elif eh <= 4.8e+132:
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(Float64(t * eh) / ew) * -0.3333333333333333))) * eh))
	tmp = 0.0
	if (eh <= -1.42e+116)
		tmp = t_1;
	elseif (eh <= 4.8e+132)
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan((((t * eh) / ew) * -0.3333333333333333))) * eh));
	tmp = 0.0;
	if (eh <= -1.42e+116)
		tmp = t_1;
	elseif (eh <= 4.8e+132)
		tmp = abs((sin(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[(t * eh), $MachinePrecision] / ew), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.42e+116], t$95$1, If[LessEqual[eh, 4.8e+132], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.4199999999999999e116 or 4.8000000000000002e132 < 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 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 40.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 40.6%

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

   12. Taylor expanded in eh around 0

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

   14. Applied rewrites 40.6%

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

   if -1.4199999999999999e116 < eh < 4.8000000000000002e132

   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

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 55.5

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

   11. Applied rewrites 55.5%

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

Alternative 9: 41.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\\ \mathbf{if}\;eh \leq -2.3 \cdot 10^{+187}:\\ \;\;\;\;\left|\frac{t\_1}{\sqrt{{t\_1}^{2} + 1}} \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (* -0.3333333333333333 (/ eh ew)) t)))
   (if (<= eh -2.3e+187)
     (fabs (* (/ t_1 (sqrt (+ (pow t_1 2.0) 1.0))) eh))
     (fabs (* (sin t) ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (-0.3333333333333333 * (eh / ew)) * t;
	double tmp;
	if (eh <= -2.3e+187) {
		tmp = fabs(((t_1 / sqrt((pow(t_1, 2.0) + 1.0))) * eh));
	} else {
		tmp = fabs((sin(t) * ew));
	}
	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 = ((-0.3333333333333333d0) * (eh / ew)) * t
    if (eh <= (-2.3d+187)) then
        tmp = abs(((t_1 / sqrt(((t_1 ** 2.0d0) + 1.0d0))) * eh))
    else
        tmp = abs((sin(t) * ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (-0.3333333333333333 * (eh / ew)) * t;
	double tmp;
	if (eh <= -2.3e+187) {
		tmp = Math.abs(((t_1 / Math.sqrt((Math.pow(t_1, 2.0) + 1.0))) * eh));
	} else {
		tmp = Math.abs((Math.sin(t) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (-0.3333333333333333 * (eh / ew)) * t
	tmp = 0
	if eh <= -2.3e+187:
		tmp = math.fabs(((t_1 / math.sqrt((math.pow(t_1, 2.0) + 1.0))) * eh))
	else:
		tmp = math.fabs((math.sin(t) * ew))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-0.3333333333333333 * Float64(eh / ew)) * t)
	tmp = 0.0
	if (eh <= -2.3e+187)
		tmp = abs(Float64(Float64(t_1 / sqrt(Float64((t_1 ^ 2.0) + 1.0))) * eh));
	else
		tmp = abs(Float64(sin(t) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (-0.3333333333333333 * (eh / ew)) * t;
	tmp = 0.0;
	if (eh <= -2.3e+187)
		tmp = abs(((t_1 / sqrt(((t_1 ^ 2.0) + 1.0))) * eh));
	else
		tmp = abs((sin(t) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(-0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[eh, -2.3e+187], N[Abs[N[(N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.30000000000000004e187

   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 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 40.0%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 54.2%

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

   13. Applied rewrites 25.8%

       \[\leadsto \left|\frac{\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t}{\sqrt{{\left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right)}^{2} + 1}} \cdot eh\right| \]

   if -2.30000000000000004e187 < eh

   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

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 43.9

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

   11. Applied rewrites 43.9%

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

Alternative 10: 41.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \left|\sin t \cdot ew\right| \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/l/ N/A

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in eh around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-sin.f64 39.8

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

11. Applied rewrites 39.8%

    \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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))))))))