Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.0s

Alternatives: 12

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $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 ew around 0

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

   1. associate-*r* N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. atan-lowering-atan.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. tan-lowering-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. *-lowering-*.f64 N/A

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

5. Simplified 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $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. 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. /-lowering-/.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. *-lowering-*.f64 98.9

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

5. Simplified 98.9%

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

Alternative 3: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. un-div-inv N/A

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

   3. clear-num N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

4. Applied egg-rr 87.3%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. atan-lowering-atan.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. tan-lowering-tan.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sin-lowering-sin.f64 97.9

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

7. Simplified 97.9%

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

Alternative 4: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

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

   2. un-div-inv N/A

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

   3. clear-num N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

4. Applied egg-rr 87.3%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. atan-lowering-atan.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. tan-lowering-tan.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sin-lowering-sin.f64 97.9

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

7. Simplified 97.9%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. atan-lowering-atan.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. tan-lowering-tan.f64 97.9

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

9. Applied egg-rr 97.9%

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

Alternative 5: 83.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := \left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{if}\;eh \leq -4.2 \cdot 10^{+61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 1.15 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{\frac{\sin t}{\sqrt{1 + \frac{eh \cdot eh}{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}}}}{\frac{1}{ew}} + t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t)))
        (t_2 (fabs (* t_1 (sin (atan (/ eh (* ew (tan t)))))))))
   (if (<= eh -4.2e+61)
     t_2
     (if (<= eh 1.15e+127)
       (fabs
        (+
         (/
          (/ (sin t) (sqrt (+ 1.0 (/ (* eh eh) (* (* t ew) (* t ew))))))
          (/ 1.0 ew))
         (* t_1 (sin (atan (/ eh (* t ew)))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = fabs((t_1 * sin(atan((eh / (ew * tan(t)))))));
	double tmp;
	if (eh <= -4.2e+61) {
		tmp = t_2;
	} else if (eh <= 1.15e+127) {
		tmp = fabs((((sin(t) / sqrt((1.0 + ((eh * eh) / ((t * ew) * (t * ew)))))) / (1.0 / ew)) + (t_1 * sin(atan((eh / (t * ew)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = eh * cos(t)
    t_2 = abs((t_1 * sin(atan((eh / (ew * tan(t)))))))
    if (eh <= (-4.2d+61)) then
        tmp = t_2
    else if (eh <= 1.15d+127) then
        tmp = abs((((sin(t) / sqrt((1.0d0 + ((eh * eh) / ((t * ew) * (t * ew)))))) / (1.0d0 / ew)) + (t_1 * sin(atan((eh / (t * ew)))))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double t_2 = Math.abs((t_1 * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	double tmp;
	if (eh <= -4.2e+61) {
		tmp = t_2;
	} else if (eh <= 1.15e+127) {
		tmp = Math.abs((((Math.sin(t) / Math.sqrt((1.0 + ((eh * eh) / ((t * ew) * (t * ew)))))) / (1.0 / ew)) + (t_1 * Math.sin(Math.atan((eh / (t * ew)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	t_2 = math.fabs((t_1 * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	tmp = 0
	if eh <= -4.2e+61:
		tmp = t_2
	elif eh <= 1.15e+127:
		tmp = math.fabs((((math.sin(t) / math.sqrt((1.0 + ((eh * eh) / ((t * ew) * (t * ew)))))) / (1.0 / ew)) + (t_1 * math.sin(math.atan((eh / (t * ew)))))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = abs(Float64(t_1 * sin(atan(Float64(eh / Float64(ew * tan(t)))))))
	tmp = 0.0
	if (eh <= -4.2e+61)
		tmp = t_2;
	elseif (eh <= 1.15e+127)
		tmp = abs(Float64(Float64(Float64(sin(t) / sqrt(Float64(1.0 + Float64(Float64(eh * eh) / Float64(Float64(t * ew) * Float64(t * ew)))))) / Float64(1.0 / ew)) + Float64(t_1 * sin(atan(Float64(eh / Float64(t * ew)))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	t_2 = abs((t_1 * sin(atan((eh / (ew * tan(t)))))));
	tmp = 0.0;
	if (eh <= -4.2e+61)
		tmp = t_2;
	elseif (eh <= 1.15e+127)
		tmp = abs((((sin(t) / sqrt((1.0 + ((eh * eh) / ((t * ew) * (t * ew)))))) / (1.0 / ew)) + (t_1 * sin(atan((eh / (t * ew)))))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -4.2e+61], t$95$2, If[LessEqual[eh, 1.15e+127], N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[N[(1.0 + N[(N[(eh * eh), $MachinePrecision] / N[(N[(t * ew), $MachinePrecision] * N[(t * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 / ew), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.2000000000000002e61 or 1.1500000000000001e127 < 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. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. atan-lowering-atan.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 90.7

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

   5. Simplified 90.7%

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

   if -4.2000000000000002e61 < eh < 1.1500000000000001e127

   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

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

      1. /-lowering-/.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. *-lowering-*.f64 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 92.9

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

   8. Simplified 92.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. remove-double-div N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 83.7%

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

Alternative 6: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ t_2 := \sqrt{1 + \frac{eh \cdot eh}{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}}\\ \mathbf{if}\;eh \leq -5.4 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.065:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{t\_2}, ew, eh \cdot \left(\cos t \cdot \frac{eh}{\left(t \cdot ew\right) \cdot t\_2}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* eh (cos t)) (sin (atan (/ eh (* ew (tan t))))))))
        (t_2 (sqrt (+ 1.0 (/ (* eh eh) (* (* t ew) (* t ew)))))))
   (if (<= eh -5.4e+21)
     t_1
     (if (<= eh 0.065)
       (fabs
        (fma (/ (sin t) t_2) ew (* eh (* (cos t) (/ eh (* (* t ew) t_2))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	double t_2 = sqrt((1.0 + ((eh * eh) / ((t * ew) * (t * ew)))));
	double tmp;
	if (eh <= -5.4e+21) {
		tmp = t_1;
	} else if (eh <= 0.065) {
		tmp = fabs(fma((sin(t) / t_2), ew, (eh * (cos(t) * (eh / ((t * ew) * t_2))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))
	t_2 = sqrt(Float64(1.0 + Float64(Float64(eh * eh) / Float64(Float64(t * ew) * Float64(t * ew)))))
	tmp = 0.0
	if (eh <= -5.4e+21)
		tmp = t_1;
	elseif (eh <= 0.065)
		tmp = abs(fma(Float64(sin(t) / t_2), ew, Float64(eh * Float64(cos(t) * Float64(eh / Float64(Float64(t * ew) * t_2))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + N[(N[(eh * eh), $MachinePrecision] / N[(N[(t * ew), $MachinePrecision] * N[(t * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -5.4e+21], t$95$1, If[LessEqual[eh, 0.065], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / t$95$2), $MachinePrecision] * ew + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[(eh / N[(N[(t * ew), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.4e21 or 0.065000000000000002 < 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. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. atan-lowering-atan.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 85.7

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

   5. Simplified 85.7%

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

   if -5.4e21 < eh < 0.065000000000000002

   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

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

      1. /-lowering-/.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. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 93.2

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

   8. Simplified 93.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 72.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.4 \cdot 10^{+21}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 0.065:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{1 + \frac{eh \cdot eh}{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}}}, ew, eh \cdot \left(\cos t \cdot \frac{eh}{\left(t \cdot ew\right) \cdot \sqrt{1 + \frac{eh \cdot eh}{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}}}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{1 + \frac{eh \cdot eh}{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}}\\ \mathbf{if}\;t \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{elif}\;t \leq 0.00037:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{t\_1}, ew, eh \cdot \left(\cos t \cdot \frac{eh}{\left(t \cdot ew\right) \cdot t\_1}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 (/ (* eh eh) (* (* t ew) (* t ew)))))))
   (if (<= t -5.1e-8)
     (fabs (* ew (sin t)))
     (if (<= t 0.00037)
       (fabs (fma eh (sin (atan (/ eh (* ew (tan t))))) (* t ew)))
       (fabs
        (fma
         (/ (sin t) t_1)
         ew
         (* eh (* (cos t) (/ eh (* (* t ew) t_1))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sqrt((1.0 + ((eh * eh) / ((t * ew) * (t * ew)))));
	double tmp;
	if (t <= -5.1e-8) {
		tmp = fabs((ew * sin(t)));
	} else if (t <= 0.00037) {
		tmp = fabs(fma(eh, sin(atan((eh / (ew * tan(t))))), (t * ew)));
	} else {
		tmp = fabs(fma((sin(t) / t_1), ew, (eh * (cos(t) * (eh / ((t * ew) * t_1))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = sqrt(Float64(1.0 + Float64(Float64(eh * eh) / Float64(Float64(t * ew) * Float64(t * ew)))))
	tmp = 0.0
	if (t <= -5.1e-8)
		tmp = abs(Float64(ew * sin(t)));
	elseif (t <= 0.00037)
		tmp = abs(fma(eh, sin(atan(Float64(eh / Float64(ew * tan(t))))), Float64(t * ew)));
	else
		tmp = abs(fma(Float64(sin(t) / t_1), ew, Float64(eh * Float64(cos(t) * Float64(eh / Float64(Float64(t * ew) * t_1))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + N[(N[(eh * eh), $MachinePrecision] / N[(N[(t * ew), $MachinePrecision] * N[(t * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -5.1e-8], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 0.00037], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / t$95$1), $MachinePrecision] * ew + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[(eh / N[(N[(t * ew), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.10000000000000001e-8

   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. cos-atan N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in eh around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 50.4

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

   7. Simplified 50.4%

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

   if -5.10000000000000001e-8 < t < 3.6999999999999999e-4

   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. Step-by-step derivation

      1. cos-atan N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. atan-lowering-atan.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 98.0

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

   7. Simplified 98.0%

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

   8. Taylor expanded in t around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. atan-lowering-atan.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.0

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

   10. Simplified 98.0%

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

   if 3.6999999999999999e-4 < 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. 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. /-lowering-/.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. *-lowering-*.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 79.6

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

   8. Simplified 79.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 61.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{elif}\;t \leq 0.00037:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{1 + \frac{eh \cdot eh}{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}}}, ew, eh \cdot \left(\cos t \cdot \frac{eh}{\left(t \cdot ew\right) \cdot \sqrt{1 + \frac{eh \cdot eh}{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}}}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.24:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -5.1e-8)
     t_1
     (if (<= t 0.24)
       (fabs (fma eh (sin (atan (/ eh (* ew (tan t))))) (* t ew)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -5.1e-8) {
		tmp = t_1;
	} else if (t <= 0.24) {
		tmp = fabs(fma(eh, sin(atan((eh / (ew * tan(t))))), (t * ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -5.1e-8)
		tmp = t_1;
	elseif (t <= 0.24)
		tmp = abs(fma(eh, sin(atan(Float64(eh / Float64(ew * tan(t))))), Float64(t * ew)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -5.1e-8], t$95$1, If[LessEqual[t, 0.24], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.10000000000000001e-8 or 0.23999999999999999 < 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. cos-atan N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in eh around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 50.6

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

   7. Simplified 50.6%

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

   if -5.10000000000000001e-8 < t < 0.23999999999999999

   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. Step-by-step derivation

      1. cos-atan N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 86.2%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. sin-lowering-sin.f64 N/A

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

      6. atan-lowering-atan.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. tan-lowering-tan.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 98.0

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

   7. Simplified 98.0%

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

   8. Taylor expanded in t around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. atan-lowering-atan.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 96.9

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

   10. Simplified 96.9%

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

Alternative 9: 67.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -8.5 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 2.8 \cdot 10^{-10}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -8.5e+61)
     t_1
     (if (<= ew 2.8e-10)
       (fabs (* (* eh (cos t)) (sin (atan (/ eh (* t ew))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -8.5e+61) {
		tmp = t_1;
	} else if (ew <= 2.8e-10) {
		tmp = fabs(((eh * cos(t)) * sin(atan((eh / (t * ew))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * sin(t)))
    if (ew <= (-8.5d+61)) then
        tmp = t_1
    else if (ew <= 2.8d-10) then
        tmp = abs(((eh * cos(t)) * sin(atan((eh / (t * ew))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.sin(t)));
	double tmp;
	if (ew <= -8.5e+61) {
		tmp = t_1;
	} else if (ew <= 2.8e-10) {
		tmp = Math.abs(((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (t * ew))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -8.5e+61:
		tmp = t_1
	elif ew <= 2.8e-10:
		tmp = math.fabs(((eh * math.cos(t)) * math.sin(math.atan((eh / (t * ew))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -8.5e+61)
		tmp = t_1;
	elseif (ew <= 2.8e-10)
		tmp = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(t * ew))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -8.5e+61)
		tmp = t_1;
	elseif (ew <= 2.8e-10)
		tmp = abs(((eh * cos(t)) * sin(atan((eh / (t * ew))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -8.5e+61], t$95$1, If[LessEqual[ew, 2.8e-10], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -8.50000000000000035e61 or 2.80000000000000015e-10 < ew

   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. Step-by-step derivation

      1. cos-atan N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 89.7%

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

   5. Taylor expanded in eh around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 68.1

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

   7. Simplified 68.1%

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

   if -8.50000000000000035e61 < ew < 2.80000000000000015e-10

   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. /-lowering-/.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. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 90.1

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

   8. Simplified 90.1%

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

   9. Taylor expanded in ew around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. cos-lowering-cos.f64 N/A

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

       5. sin-lowering-sin.f64 N/A

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

       6. atan-lowering-atan.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 72.5

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

   11. Simplified 72.5%

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

Alternative 10: 58.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -9.6 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.45 \cdot 10^{-8}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -9.6e-85) t_1 (if (<= ew 1.45e-8) (fabs eh) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -9.6e-85) {
		tmp = t_1;
	} else if (ew <= 1.45e-8) {
		tmp = fabs(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((ew * sin(t)))
    if (ew <= (-9.6d-85)) then
        tmp = t_1
    else if (ew <= 1.45d-8) then
        tmp = abs(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((ew * Math.sin(t)));
	double tmp;
	if (ew <= -9.6e-85) {
		tmp = t_1;
	} else if (ew <= 1.45e-8) {
		tmp = Math.abs(eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -9.6e-85:
		tmp = t_1
	elif ew <= 1.45e-8:
		tmp = math.fabs(eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -9.6e-85)
		tmp = t_1;
	elseif (ew <= 1.45e-8)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -9.6e-85)
		tmp = t_1;
	elseif (ew <= 1.45e-8)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -9.6e-85], t$95$1, If[LessEqual[ew, 1.45e-8], N[Abs[eh], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -9.6000000000000002e-85 or 1.4500000000000001e-8 < ew

   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. Step-by-step derivation

      1. cos-atan N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 90.4%

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

   5. Taylor expanded in eh around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 64.0

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

   7. Simplified 64.0%

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

   if -9.6000000000000002e-85 < ew < 1.4500000000000001e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. atan-lowering-atan.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. tan-lowering-tan.f64 56.3

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

   5. Simplified 56.3%

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

      1. sin-atan N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}}\right| \]

      2. div-inv N/A

         \[\leadsto \left|eh \cdot \frac{\color{blue}{eh \cdot \frac{1}{ew \cdot \tan t}}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right| \]

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. associate-/r* N/A

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

      7. div-inv N/A

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

      8. associate-/r* N/A

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

      9. div-inv N/A

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

      10. swap-sqr N/A

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

      11. inv-pow N/A

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

      12. inv-pow N/A

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

   7. Applied egg-rr 3.9%

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

   8. Taylor expanded in eh around inf

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

   10. Simplified 56.5%

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

Alternative 11: 43.2% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{-164}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 0.05:\\ \;\;\;\;\left|t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -9e-164) (fabs eh) (if (<= eh 0.05) (fabs (* t ew)) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -9e-164) {
		tmp = fabs(eh);
	} else if (eh <= 0.05) {
		tmp = fabs((t * ew));
	} else {
		tmp = fabs(eh);
	}
	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) :: tmp
    if (eh <= (-9d-164)) then
        tmp = abs(eh)
    else if (eh <= 0.05d0) then
        tmp = abs((t * ew))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -9e-164) {
		tmp = Math.abs(eh);
	} else if (eh <= 0.05) {
		tmp = Math.abs((t * ew));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -9e-164:
		tmp = math.fabs(eh)
	elif eh <= 0.05:
		tmp = math.fabs((t * ew))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -9e-164)
		tmp = abs(eh);
	elseif (eh <= 0.05)
		tmp = abs(Float64(t * ew));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -9e-164)
		tmp = abs(eh);
	elseif (eh <= 0.05)
		tmp = abs((t * ew));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -9e-164], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 0.05], N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999995e-164 or 0.050000000000000003 < 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. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. atan-lowering-atan.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. tan-lowering-tan.f64 49.1

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

   5. Simplified 49.1%

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

      1. sin-atan N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}}\right| \]

      2. div-inv N/A

         \[\leadsto \left|eh \cdot \frac{\color{blue}{eh \cdot \frac{1}{ew \cdot \tan t}}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right| \]

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. associate-/r* N/A

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

      7. div-inv N/A

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

      8. associate-/r* N/A

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

      9. div-inv N/A

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

      10. swap-sqr N/A

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

      11. inv-pow N/A

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

      12. inv-pow N/A

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

   7. Applied egg-rr 8.3%

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

   8. Taylor expanded in eh around inf

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

   10. Simplified 49.4%

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

   if -8.9999999999999995e-164 < eh < 0.050000000000000003

   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. Step-by-step derivation

      1. cos-atan N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 78.0%

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

   5. Taylor expanded in eh around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 76.6

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

   7. Simplified 76.6%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 39.1

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

   10. Simplified 39.1%

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

Alternative 12: 42.4% accurate, 290.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sin-lowering-sin.f64 N/A

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

   3. atan-lowering-atan.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. tan-lowering-tan.f64 39.4

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

5. Simplified 39.4%

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

   1. sin-atan N/A

      \[\leadsto \left|eh \cdot \color{blue}{\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}}\right| \]

   2. div-inv N/A

      \[\leadsto \left|eh \cdot \frac{\color{blue}{eh \cdot \frac{1}{ew \cdot \tan t}}}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right| \]

   3. associate-/l* N/A

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

   4. associate-/r* N/A

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

   5. associate-/r* N/A

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

   6. associate-/r* N/A

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

   7. div-inv N/A

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

   8. associate-/r* N/A

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

   9. div-inv N/A

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

   10. swap-sqr N/A

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

   11. inv-pow N/A

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

   12. inv-pow N/A

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

7. Applied egg-rr 7.5%

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

8. Taylor expanded in eh around inf

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

10. Simplified 39.9%

    \[\leadsto \left|\color{blue}{eh}\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(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))))))))