Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.9s

Alternatives: 10

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

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

FPCore

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

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 * (cos(t_1) * sin(t)))));
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * tan(t))))
	return abs(fma(eh, Float64(cos(t) * sin(t_1)), Float64(ew * Float64(cos(t_1) * sin(t)))))
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[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(ew * N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t], $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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in ew around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-atan.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.8%

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

Alternative 2: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Cos[t], $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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.4

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

5. Applied rewrites 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

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

Alternative 3: 98.4% accurate, 1.6× speedup

Math

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

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(eh * Float64(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[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $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

4. Applied rewrites 89.3%

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

5. Taylor expanded in eh around 0

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

   1. lower-sin.f64 97.7

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

7. Applied rewrites 97.7%

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

Alternative 4: 89.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := \frac{eh}{ew \cdot t}\\ \mathbf{if}\;eh \leq 9.5 \cdot 10^{+64}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}}, ew, t\_1 \cdot \sin \tan^{-1} t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))) (t_2 (/ eh (* ew t))))
   (if (<= eh 9.5e+64)
     (fabs
      (fma (/ (sin t) (sqrt (fma t_2 t_2 1.0))) ew (* t_1 (sin (atan t_2)))))
     (fabs (* (sin (atan (/ eh (* ew (tan t))))) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = eh / (ew * t);
	double tmp;
	if (eh <= 9.5e+64) {
		tmp = fabs(fma((sin(t) / sqrt(fma(t_2, t_2, 1.0))), ew, (t_1 * sin(atan(t_2)))));
	} else {
		tmp = fabs((sin(atan((eh / (ew * tan(t))))) * t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = Float64(eh / Float64(ew * t))
	tmp = 0.0
	if (eh <= 9.5e+64)
		tmp = abs(fma(Float64(sin(t) / sqrt(fma(t_2, t_2, 1.0))), ew, Float64(t_1 * sin(atan(t_2)))));
	else
		tmp = abs(Float64(sin(atan(Float64(eh / Float64(ew * tan(t))))) * t_1));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, 9.5e+64], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(t$95$1 * N[Sin[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < 9.50000000000000028e64

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 86.9

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

   10. Applied rewrites 86.9%

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

   if 9.50000000000000028e64 < 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-/.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. lower-*.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. lower-tan.f64 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 9.5 \cdot 10^{+64}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \mathbf{if}\;eh \leq -1.1 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{-239}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin (atan (/ eh (* ew (tan t))))) (* eh (cos t))))))
   (if (<= eh -1.1e-26) t_1 (if (<= eh 2.5e-239) (fabs (* ew (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan((eh / (ew * tan(t))))) * (eh * cos(t))));
	double tmp;
	if (eh <= -1.1e-26) {
		tmp = t_1;
	} else if (eh <= 2.5e-239) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(atan((eh / (ew * tan(t))))) * (eh * cos(t))))
    if (eh <= (-1.1d-26)) then
        tmp = t_1
    else if (eh <= 2.5d-239) then
        tmp = abs((ew * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(Math.atan((eh / (ew * Math.tan(t))))) * (eh * Math.cos(t))));
	double tmp;
	if (eh <= -1.1e-26) {
		tmp = t_1;
	} else if (eh <= 2.5e-239) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan((eh / (ew * math.tan(t))))) * (eh * math.cos(t))))
	tmp = 0
	if eh <= -1.1e-26:
		tmp = t_1
	elif eh <= 2.5e-239:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(eh / Float64(ew * tan(t))))) * Float64(eh * cos(t))))
	tmp = 0.0
	if (eh <= -1.1e-26)
		tmp = t_1;
	elseif (eh <= 2.5e-239)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan((eh / (ew * tan(t))))) * (eh * cos(t))));
	tmp = 0.0;
	if (eh <= -1.1e-26)
		tmp = t_1;
	elseif (eh <= 2.5e-239)
		tmp = abs((ew * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.1e-26], t$95$1, If[LessEqual[eh, 2.5e-239], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.1e-26 or 2.5e-239 < 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-/.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. lower-*.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. lower-tan.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if -1.1e-26 < eh < 2.5e-239

   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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 80.8

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

   7. Applied rewrites 80.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.1 \cdot 10^{-26}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{-239}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq 9.5 \cdot 10^{+64}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin t, ew, t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= eh 9.5e+64)
     (fabs (fma (sin t) ew (* t_1 (sin (atan (/ eh (* ew t)))))))
     (fabs (* (sin (atan (/ eh (* ew (tan t))))) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (eh <= 9.5e+64) {
		tmp = fabs(fma(sin(t), ew, (t_1 * sin(atan((eh / (ew * t)))))));
	} else {
		tmp = fabs((sin(atan((eh / (ew * tan(t))))) * t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= 9.5e+64)
		tmp = abs(fma(sin(t), ew, Float64(t_1 * sin(atan(Float64(eh / Float64(ew * t)))))));
	else
		tmp = abs(Float64(sin(atan(Float64(eh / Float64(ew * tan(t))))) * t_1));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, 9.5e+64], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew + N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < 9.50000000000000028e64

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in eh around 0

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

      1. lower-sin.f64 97.7

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

   10. Applied rewrites 97.7%

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

   11. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 86.0

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

   13. Applied rewrites 86.0%

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

   if 9.50000000000000028e64 < 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-/.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. lower-*.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. lower-tan.f64 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 9.5 \cdot 10^{+64}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin t, ew, \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|-eh\right|\\ \mathbf{if}\;eh \leq -1.15 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 9.5 \cdot 10^{+20}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (- eh))))
   (if (<= eh -1.15e-26) t_1 (if (<= eh 9.5e+20) (fabs (* ew (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(-eh);
	double tmp;
	if (eh <= -1.15e-26) {
		tmp = t_1;
	} else if (eh <= 9.5e+20) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(-eh)
    if (eh <= (-1.15d-26)) then
        tmp = t_1
    else if (eh <= 9.5d+20) then
        tmp = abs((ew * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(-eh);
	double tmp;
	if (eh <= -1.15e-26) {
		tmp = t_1;
	} else if (eh <= 9.5e+20) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(-eh)
	tmp = 0
	if eh <= -1.15e-26:
		tmp = t_1
	elif eh <= 9.5e+20:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(-eh))
	tmp = 0.0
	if (eh <= -1.15e-26)
		tmp = t_1;
	elseif (eh <= 9.5e+20)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(-eh);
	tmp = 0.0;
	if (eh <= -1.15e-26)
		tmp = t_1;
	elseif (eh <= 9.5e+20)
		tmp = abs((ew * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[(-eh)], $MachinePrecision]}, If[LessEqual[eh, -1.15e-26], t$95$1, If[LessEqual[eh, 9.5e+20], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.15000000000000004e-26 or 9.5e20 < 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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-tan.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 9.1%

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

   8. Taylor expanded in eh around -inf

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

   10. Applied rewrites 60.0%

       \[\leadsto \left|-eh\right| \]

   if -1.15000000000000004e-26 < eh < 9.5e20

   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

   4. Applied rewrites 35.2%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 64.2

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

   7. Applied rewrites 64.2%

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

Alternative 8: 44.5% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1 \cdot 10^{+143}:\\ \;\;\;\;\left|t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.008333333333333333, ew \cdot \left(t \cdot t\right), ew \cdot -0.16666666666666666\right), ew\right)\right|\\ \mathbf{elif}\;ew \leq 1.15 \cdot 10^{+191}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1e+143)
   (fabs
    (*
     t
     (fma
      (* t t)
      (fma 0.008333333333333333 (* ew (* t t)) (* ew -0.16666666666666666))
      ew)))
   (if (<= ew 1.15e+191) (fabs (- eh)) (fabs (* ew t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1e+143) {
		tmp = fabs((t * fma((t * t), fma(0.008333333333333333, (ew * (t * t)), (ew * -0.16666666666666666)), ew)));
	} else if (ew <= 1.15e+191) {
		tmp = fabs(-eh);
	} else {
		tmp = fabs((ew * t));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1e+143)
		tmp = abs(Float64(t * fma(Float64(t * t), fma(0.008333333333333333, Float64(ew * Float64(t * t)), Float64(ew * -0.16666666666666666)), ew)));
	elseif (ew <= 1.15e+191)
		tmp = abs(Float64(-eh));
	else
		tmp = abs(Float64(ew * t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1e+143], N[Abs[N[(t * N[(N[(t * t), $MachinePrecision] * N[(0.008333333333333333 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(ew * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.15e+191], N[Abs[(-eh)], $MachinePrecision], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1e143

   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

   4. Applied rewrites 25.1%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 90.6

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|t \cdot \color{blue}{\left(ew + {t}^{2} \cdot \left(\frac{-1}{6} \cdot ew + \frac{1}{120} \cdot \left(ew \cdot {t}^{2}\right)\right)\right)}\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 49.1%

       \[\leadsto \left|t \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.008333333333333333, ew \cdot \left(t \cdot t\right), -0.16666666666666666 \cdot ew\right), ew\right)}\right| \]

   if -1e143 < ew < 1.15e191

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-tan.f64 49.9

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

   5. Applied rewrites 49.9%

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in eh around -inf

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

   10. Applied rewrites 50.3%

       \[\leadsto \left|-eh\right| \]

   if 1.15e191 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 22.9%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 79.3

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 49.3%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1 \cdot 10^{+143}:\\ \;\;\;\;\left|t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.008333333333333333, ew \cdot \left(t \cdot t\right), ew \cdot -0.16666666666666666\right), ew\right)\right|\\ \mathbf{elif}\;ew \leq 1.15 \cdot 10^{+191}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.5% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot t\right|\\ \mathbf{if}\;ew \leq -2.75 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.15 \cdot 10^{+191}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew t))))
   (if (<= ew -2.75e+150) t_1 (if (<= ew 1.15e+191) (fabs (- eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * t));
	double tmp;
	if (ew <= -2.75e+150) {
		tmp = t_1;
	} else if (ew <= 1.15e+191) {
		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 * t))
    if (ew <= (-2.75d+150)) then
        tmp = t_1
    else if (ew <= 1.15d+191) 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 * t));
	double tmp;
	if (ew <= -2.75e+150) {
		tmp = t_1;
	} else if (ew <= 1.15e+191) {
		tmp = Math.abs(-eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * t))
	tmp = 0
	if ew <= -2.75e+150:
		tmp = t_1
	elif ew <= 1.15e+191:
		tmp = math.fabs(-eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * t))
	tmp = 0.0
	if (ew <= -2.75e+150)
		tmp = t_1;
	elseif (ew <= 1.15e+191)
		tmp = abs(Float64(-eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * t));
	tmp = 0.0;
	if (ew <= -2.75e+150)
		tmp = t_1;
	elseif (ew <= 1.15e+191)
		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 * t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.75e+150], t$95$1, If[LessEqual[ew, 1.15e+191], N[Abs[(-eh)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.75000000000000008e150 or 1.15e191 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 25.7%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 87.7

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 50.0%

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

   if -2.75000000000000008e150 < ew < 1.15e191

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-tan.f64 49.7

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

   5. Applied rewrites 49.7%

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

   7. Applied rewrites 8.0%

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

   8. Taylor expanded in eh around -inf

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

   10. Applied rewrites 50.1%

       \[\leadsto \left|-eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 41.9% accurate, 174.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-atan.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-tan.f64 43.0

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

5. Applied rewrites 43.0%

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

7. Applied rewrites 8.2%

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

8. Taylor expanded in eh around -inf

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

10. Applied rewrites 43.4%

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

Reproduce

herbie shell --seed 2024233 
(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))))))))