Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.9s

Alternatives: 8

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. lower-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. lower-*.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. lower-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. lower-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. lower-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. lower-/.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. lower-*.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. lower-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. lower-*.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: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left|t\_1 \cdot \sin t\_2 + \left(ew \cdot \sin t\right) \cdot \cos t\_2\right| \leq 10^{-165}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\cos t}{ew}, \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}, ew, t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))) (t_2 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (fabs (+ (* t_1 (sin t_2)) (* (* ew (sin t)) (cos t_2)))) 1e-165)
     (fabs (* ew (fma eh (/ (cos t) ew) (sin t))))
     (fabs
      (fma
       (/ (sin t) (sqrt (fma (* eh eh) (pow (* ew (tan t)) -2.0) 1.0)))
       ew
       t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (fabs(((t_1 * sin(t_2)) + ((ew * sin(t)) * cos(t_2)))) <= 1e-165) {
		tmp = fabs((ew * fma(eh, (cos(t) / ew), sin(t))));
	} else {
		tmp = fabs(fma((sin(t) / sqrt(fma((eh * eh), pow((ew * tan(t)), -2.0), 1.0))), ew, t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (abs(Float64(Float64(t_1 * sin(t_2)) + Float64(Float64(ew * sin(t)) * cos(t_2)))) <= 1e-165)
		tmp = abs(Float64(ew * fma(eh, Float64(cos(t) / ew), sin(t))));
	else
		tmp = abs(fma(Float64(sin(t) / sqrt(fma(Float64(eh * eh), (Float64(ew * tan(t)) ^ -2.0), 1.0))), ew, 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[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e-165], N[Abs[N[(ew * N[(eh * N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[N[(N[(eh * eh), $MachinePrecision] * N[Power[N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))) < 1e-165

   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 egg-rr 0.0%

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

   5. Taylor expanded in eh around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 0.0

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

   7. Simplified 0.0%

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

   8. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 93.6

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

   10. Simplified 93.6%

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

   if 1e-165 < (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))))

   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 egg-rr 47.0%

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

   5. Taylor expanded in eh around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 49.0

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

   7. Simplified 49.0%

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

   9. Applied egg-rr 97.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \leq 10^{-165}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\cos t}{ew}, \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}, ew, eh \cdot \cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% 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. 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 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. Add Preprocessing

Alternative 4: 93.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \cos t\right|\\ \mathbf{if}\;eh \leq -3.5 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 7 \cdot 10^{+110}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\cos t}{ew}, \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (cos t)))))
   (if (<= eh -3.5e+98)
     t_1
     (if (<= eh 7e+110) (fabs (* ew (fma eh (/ (cos t) ew) (sin t)))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * cos(t)));
	double tmp;
	if (eh <= -3.5e+98) {
		tmp = t_1;
	} else if (eh <= 7e+110) {
		tmp = fabs((ew * fma(eh, (cos(t) / ew), sin(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * cos(t)))
	tmp = 0.0
	if (eh <= -3.5e+98)
		tmp = t_1;
	elseif (eh <= 7e+110)
		tmp = abs(Float64(ew * fma(eh, Float64(cos(t) / ew), sin(t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.5e+98], t$95$1, If[LessEqual[eh, 7e+110], N[Abs[N[(ew * N[(eh * N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.5e98 or 6.9999999999999998e110 < eh

   1. Initial program 99.9%

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

   4. Applied egg-rr 4.5%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 94.6

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

   7. Simplified 94.6%

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

   if -3.5e98 < eh < 6.9999999999999998e110

   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 egg-rr 62.8%

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

   5. Taylor expanded in eh around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 64.2

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

   7. Simplified 64.2%

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

   8. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 96.4

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

   10. Simplified 96.4%

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

Alternative 5: 74.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \cos t\right|\\ \mathbf{if}\;eh \leq -8.5 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.35 \cdot 10^{-71}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (cos t)))))
   (if (<= eh -8.5e-35) t_1 (if (<= eh 2.35e-71) (fabs (* ew (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * cos(t)));
	double tmp;
	if (eh <= -8.5e-35) {
		tmp = t_1;
	} else if (eh <= 2.35e-71) {
		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 * cos(t)))
    if (eh <= (-8.5d-35)) then
        tmp = t_1
    else if (eh <= 2.35d-71) 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 * Math.cos(t)));
	double tmp;
	if (eh <= -8.5e-35) {
		tmp = t_1;
	} else if (eh <= 2.35e-71) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.cos(t)))
	tmp = 0
	if eh <= -8.5e-35:
		tmp = t_1
	elif eh <= 2.35e-71:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * cos(t)))
	tmp = 0.0
	if (eh <= -8.5e-35)
		tmp = t_1;
	elseif (eh <= 2.35e-71)
		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 * cos(t)));
	tmp = 0.0;
	if (eh <= -8.5e-35)
		tmp = t_1;
	elseif (eh <= 2.35e-71)
		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[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -8.5e-35], t$95$1, If[LessEqual[eh, 2.35e-71], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.5000000000000001e-35 or 2.34999999999999998e-71 < 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

   4. Applied egg-rr 27.3%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 84.7

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

   7. Simplified 84.7%

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

   if -8.5000000000000001e-35 < eh < 2.34999999999999998e-71

   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 egg-rr 65.0%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 75.1

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

   7. Simplified 75.1%

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

Alternative 6: 63.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \cos t\right|\\ \mathbf{if}\;eh \leq -6.5 \cdot 10^{-187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 5.8 \cdot 10^{-147}:\\ \;\;\;\;\left|t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (cos t)))))
   (if (<= eh -6.5e-187) t_1 (if (<= eh 5.8e-147) (fabs (* t ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * cos(t)));
	double tmp;
	if (eh <= -6.5e-187) {
		tmp = t_1;
	} else if (eh <= 5.8e-147) {
		tmp = fabs((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((eh * cos(t)))
    if (eh <= (-6.5d-187)) then
        tmp = t_1
    else if (eh <= 5.8d-147) then
        tmp = abs((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((eh * Math.cos(t)));
	double tmp;
	if (eh <= -6.5e-187) {
		tmp = t_1;
	} else if (eh <= 5.8e-147) {
		tmp = Math.abs((t * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.cos(t)))
	tmp = 0
	if eh <= -6.5e-187:
		tmp = t_1
	elif eh <= 5.8e-147:
		tmp = math.fabs((t * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * cos(t)))
	tmp = 0.0
	if (eh <= -6.5e-187)
		tmp = t_1;
	elseif (eh <= 5.8e-147)
		tmp = abs(Float64(t * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * cos(t)));
	tmp = 0.0;
	if (eh <= -6.5e-187)
		tmp = t_1;
	elseif (eh <= 5.8e-147)
		tmp = abs((t * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -6.5e-187], t$95$1, If[LessEqual[eh, 5.8e-147], N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.49999999999999983e-187 or 5.8000000000000002e-147 < 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

   4. Applied egg-rr 38.8%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 71.6

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

   7. Simplified 71.6%

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

   if -6.49999999999999983e-187 < eh < 5.8000000000000002e-147

   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 egg-rr 58.8%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 80.9

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

   7. Simplified 80.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{\frac{1}{2} \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(ew + \frac{1}{2} \cdot \left(-1 \cdot \frac{{eh}^{2}}{ew} - \frac{-1}{6} \cdot \frac{{eh}^{2}}{ew}\right)\right)}{t}}\right| \]

   10. Simplified 34.6%

       \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.5, \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334, ew\right), \frac{0.5 \cdot \left(eh \cdot eh\right)}{ew}\right)}{t}}\right| \]

   11. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 53.2

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

   13. Simplified 53.2%

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

Alternative 7: 45.9% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{-187}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 10^{-146}:\\ \;\;\;\;\left|t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -9e-187) (fabs eh) (if (<= eh 1e-146) (fabs (* t ew)) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -9e-187) {
		tmp = fabs(eh);
	} else if (eh <= 1e-146) {
		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-187)) then
        tmp = abs(eh)
    else if (eh <= 1d-146) 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-187) {
		tmp = Math.abs(eh);
	} else if (eh <= 1e-146) {
		tmp = Math.abs((t * ew));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -9e-187:
		tmp = math.fabs(eh)
	elif eh <= 1e-146:
		tmp = math.fabs((t * ew))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -9e-187)
		tmp = abs(eh);
	elseif (eh <= 1e-146)
		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-187)
		tmp = abs(eh);
	elseif (eh <= 1e-146)
		tmp = abs((t * ew));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -9e-187], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 1e-146], N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999996e-187 or 1.00000000000000003e-146 < 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

   4. Applied egg-rr 38.8%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 71.6

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

   7. Simplified 71.6%

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

   8. Taylor expanded in t around 0

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

   10. Simplified 52.8%

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

       1. *-rgt-identity 52.8

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

   12. Applied egg-rr 52.8%

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

   if -8.9999999999999996e-187 < eh < 1.00000000000000003e-146

   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 egg-rr 58.8%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 80.9

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

   7. Simplified 80.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{\frac{1}{2} \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(ew + \frac{1}{2} \cdot \left(-1 \cdot \frac{{eh}^{2}}{ew} - \frac{-1}{6} \cdot \frac{{eh}^{2}}{ew}\right)\right)}{t}}\right| \]

   10. Simplified 34.6%

       \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.5, \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334, ew\right), \frac{0.5 \cdot \left(eh \cdot eh\right)}{ew}\right)}{t}}\right| \]

   11. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 53.2

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

   13. Simplified 53.2%

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

Alternative 8: 43.0% 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

4. Applied egg-rr 42.8%

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

5. Taylor expanded in eh around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 61.7

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

7. Simplified 61.7%

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

8. Taylor expanded in t around 0

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

10. Simplified 45.1%

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

    1. *-rgt-identity 45.1

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

12. Applied egg-rr 45.1%

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

Reproduce

herbie shell --seed 2024208 
(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))))))))