Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((-eh * Math.tan(t)) / ew));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((-eh * Math.tan(t)) / ew));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = eh * tan(t)
    code = abs(((cos(atan((t_1 / ew))) * (ew * cos(t))) - ((eh * sin(t)) * sin(atan((t_1 / -ew))))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.tan(t);
	return Math.abs(((Math.cos(Math.atan((t_1 / ew))) * (ew * Math.cos(t))) - ((eh * Math.sin(t)) * Math.sin(Math.atan((t_1 / -ew))))));
}

Python

def code(eh, ew, t):
	t_1 = eh * math.tan(t)
	return math.fabs(((math.cos(math.atan((t_1 / ew))) * (ew * math.cos(t))) - ((eh * math.sin(t)) * math.sin(math.atan((t_1 / -ew))))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = eh * tan(t);
	tmp = abs(((cos(atan((t_1 / ew))) * (ew * cos(t))) - ((eh * sin(t)) * sin(atan((t_1 / -ew))))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. add-sqr-sqrt 51.1%

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

   2. sqrt-unprod 93.5%

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

   3. sqr-neg 93.5%

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

   4. sqrt-unprod 48.8%

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

   5. add-sqr-sqrt 99.9%

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

   6. pow1 99.9%

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

4. Applied egg-rr 99.9%

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

   1. unpow1 99.9%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * (Math.tan(t) / ew);
	return Math.abs((((ew * Math.cos(t)) / Math.hypot(1.0, t_1)) + (eh * (Math.sin(t) * Math.sin(Math.atan(t_1))))));
}

Python

def code(eh, ew, t):
	t_1 = eh * (math.tan(t) / ew)
	return math.fabs((((ew * math.cos(t)) / math.hypot(1.0, t_1)) + (eh * (math.sin(t) * math.sin(math.atan(t_1))))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = eh * (tan(t) / ew);
	tmp = abs((((ew * cos(t)) / hypot(1.0, t_1)) + (eh * (sin(t) * sin(atan(t_1))))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l* 99.8%

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

   6. add-sqr-sqrt 51.1%

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

   7. sqrt-unprod 93.1%

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

   8. sqr-neg 93.1%

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

   9. sqrt-unprod 48.8%

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

   10. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 3: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;ew \leq -4.1 \cdot 10^{-155} \lor \neg \left(ew \leq 1.12 \cdot 10^{-170}\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t - eh \cdot \left(\sin t \cdot \frac{\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} t\_1\right) + \frac{ew}{\mathsf{hypot}\left(1, t\_1\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))))
   (if (or (<= ew -4.1e-155) (not (<= ew 1.12e-170)))
     (fabs
      (*
       ew
       (-
        (cos t)
        (* eh (* (sin t) (/ (sin (atan (* eh (/ (tan t) (- ew))))) ew))))))
     (fabs (+ (* eh (* (sin t) (sin (atan t_1)))) (/ ew (hypot 1.0 t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double tmp;
	if ((ew <= -4.1e-155) || !(ew <= 1.12e-170)) {
		tmp = fabs((ew * (cos(t) - (eh * (sin(t) * (sin(atan((eh * (tan(t) / -ew)))) / ew))))));
	} else {
		tmp = fabs(((eh * (sin(t) * sin(atan(t_1)))) + (ew / hypot(1.0, t_1))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * (Math.tan(t) / ew);
	double tmp;
	if ((ew <= -4.1e-155) || !(ew <= 1.12e-170)) {
		tmp = Math.abs((ew * (Math.cos(t) - (eh * (Math.sin(t) * (Math.sin(Math.atan((eh * (Math.tan(t) / -ew)))) / ew))))));
	} else {
		tmp = Math.abs(((eh * (Math.sin(t) * Math.sin(Math.atan(t_1)))) + (ew / Math.hypot(1.0, t_1))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * (math.tan(t) / ew)
	tmp = 0
	if (ew <= -4.1e-155) or not (ew <= 1.12e-170):
		tmp = math.fabs((ew * (math.cos(t) - (eh * (math.sin(t) * (math.sin(math.atan((eh * (math.tan(t) / -ew)))) / ew))))))
	else:
		tmp = math.fabs(((eh * (math.sin(t) * math.sin(math.atan(t_1)))) + (ew / math.hypot(1.0, t_1))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if ((ew <= -4.1e-155) || !(ew <= 1.12e-170))
		tmp = abs(Float64(ew * Float64(cos(t) - Float64(eh * Float64(sin(t) * Float64(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))) / ew))))));
	else
		tmp = abs(Float64(Float64(eh * Float64(sin(t) * sin(atan(t_1)))) + Float64(ew / hypot(1.0, t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * (tan(t) / ew);
	tmp = 0.0;
	if ((ew <= -4.1e-155) || ~((ew <= 1.12e-170)))
		tmp = abs((ew * (cos(t) - (eh * (sin(t) * (sin(atan((eh * (tan(t) / -ew)))) / ew))))));
	else
		tmp = abs(((eh * (sin(t) * sin(atan(t_1)))) + (ew / hypot(1.0, t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -4.1e-155], N[Not[LessEqual[ew, 1.12e-170]], $MachinePrecision]], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ew / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.0999999999999998e-155 or 1.12000000000000009e-170 < ew

   1. Initial program 99.9%

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

      1. add-cbrt-cube 55.7%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      2. pow3 55.7%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. Applied egg-rr 57.2%

      \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right)}^{3}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf 98.3%

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

   7. Simplified 98.3%

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

   if -4.0999999999999998e-155 < ew < 1.12000000000000009e-170

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. cos-atan 99.8%

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

      3. un-div-inv 99.8%

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

      4. hypot-1-def 99.9%

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

      5. associate-/l* 99.9%

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

      6. add-sqr-sqrt 52.3%

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

      7. sqrt-unprod 99.9%

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

      8. sqr-neg 99.9%

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

      9. sqrt-unprod 47.5%

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

      10. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 97.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.1 \cdot 10^{-155} \lor \neg \left(ew \leq 1.12 \cdot 10^{-170}\right):\\ \;\;\;\;\left|ew \cdot \left(\cos t - eh \cdot \left(\sin t \cdot \frac{\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right) + \frac{ew}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -6 \cdot 10^{+203}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - eh \cdot \left(\sin t \cdot \frac{\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)}{ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -6e+203)
   (fabs (* (* eh (sin t)) (sin (atan (/ (* t (- eh)) ew)))))
   (fabs
    (*
     ew
     (-
      (cos t)
      (* eh (* (sin t) (/ (sin (atan (* eh (/ (tan t) (- ew))))) ew))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -6e+203) {
		tmp = fabs(((eh * sin(t)) * sin(atan(((t * -eh) / ew)))));
	} else {
		tmp = fabs((ew * (cos(t) - (eh * (sin(t) * (sin(atan((eh * (tan(t) / -ew)))) / ew))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-6d+203)) then
        tmp = abs(((eh * sin(t)) * sin(atan(((t * -eh) / ew)))))
    else
        tmp = abs((ew * (cos(t) - (eh * (sin(t) * (sin(atan((eh * (tan(t) / -ew)))) / ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -6e+203) {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan(((t * -eh) / ew)))));
	} else {
		tmp = Math.abs((ew * (Math.cos(t) - (eh * (Math.sin(t) * (Math.sin(Math.atan((eh * (Math.tan(t) / -ew)))) / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -6e+203:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan(((t * -eh) / ew)))))
	else:
		tmp = math.fabs((ew * (math.cos(t) - (eh * (math.sin(t) * (math.sin(math.atan((eh * (math.tan(t) / -ew)))) / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -6e+203)
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * Float64(-eh)) / ew)))));
	else
		tmp = abs(Float64(ew * Float64(cos(t) - Float64(eh * Float64(sin(t) * Float64(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -6e+203)
		tmp = abs(((eh * sin(t)) * sin(atan(((t * -eh) / ew)))));
	else
		tmp = abs((ew * (cos(t) - (eh * (sin(t) * (sin(atan((eh * (tan(t) / -ew)))) / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -6e+203], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.9999999999999999e203

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0 88.3%

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

      1. mul-1-neg 88.3%

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

      2. associate-*r* 88.3%

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

      3. distribute-lft-neg-in 88.3%

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

      4. distribute-rgt-neg-in 88.3%

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

      5. mul-1-neg 88.3%

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

      6. associate-*r/ 88.2%

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

      7. distribute-rgt-neg-in 88.2%

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

      8. distribute-neg-frac 88.2%

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

   5. Simplified 88.2%

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

   6. Taylor expanded in t around 0 88.3%

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

      1. associate-*r/ 88.3%

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

      2. associate-*r* 88.3%

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

      3. neg-mul-1 88.3%

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

   8. Simplified 88.3%

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

   if -5.9999999999999999e203 < eh

   1. Initial program 99.9%

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

      1. add-cbrt-cube 59.4%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      2. pow3 59.4%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. Applied egg-rr 60.2%

      \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right)}^{3}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf 93.9%

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

   7. Simplified 93.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6 \cdot 10^{+203}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t - eh \cdot \left(\sin t \cdot \frac{\sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{+54} \lor \neg \left(eh \leq 1.3 \cdot 10^{+53}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2e+54) (not (<= eh 1.3e+53)))
   (fabs (* (* eh (sin t)) (sin (atan (/ (* t (- eh)) ew)))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2e+54) || !(eh <= 1.3e+53)) {
		tmp = fabs(((eh * sin(t)) * sin(atan(((t * -eh) / ew)))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	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 <= (-2d+54)) .or. (.not. (eh <= 1.3d+53))) then
        tmp = abs(((eh * sin(t)) * sin(atan(((t * -eh) / ew)))))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2e+54) || !(eh <= 1.3e+53)) {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan(((t * -eh) / ew)))));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -2e+54) or not (eh <= 1.3e+53):
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(math.atan(((t * -eh) / ew)))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2e+54) || !(eh <= 1.3e+53))
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(t * Float64(-eh)) / ew)))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -2e+54) || ~((eh <= 1.3e+53)))
		tmp = abs(((eh * sin(t)) * sin(atan(((t * -eh) / ew)))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2e+54], N[Not[LessEqual[eh, 1.3e+53]], $MachinePrecision]], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.0000000000000002e54 or 1.29999999999999999e53 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around 0 71.9%

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

      1. mul-1-neg 71.9%

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

      2. associate-*r* 71.9%

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

      3. distribute-lft-neg-in 71.9%

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

      4. distribute-rgt-neg-in 71.9%

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

      5. mul-1-neg 71.9%

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

      6. associate-*r/ 71.9%

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

      7. distribute-rgt-neg-in 71.9%

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

      8. distribute-neg-frac 71.9%

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

   5. Simplified 71.9%

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

   6. Taylor expanded in t around 0 72.0%

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

      1. associate-*r/ 72.0%

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

      2. associate-*r* 72.0%

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

      3. neg-mul-1 72.0%

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

   8. Simplified 72.0%

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

   if -2.0000000000000002e54 < eh < 1.29999999999999999e53

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 50.9%

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

      2. pow2 50.9%

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

   4. Applied egg-rr 50.9%

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

   5. Taylor expanded in ew around inf 83.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{+54} \lor \neg \left(eh \leq 1.3 \cdot 10^{+53}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{t \cdot \left(-eh\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-5} \lor \neg \left(t \leq 0.005\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{t}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -4.6e-5) (not (<= t 0.005)))
   (fabs (* ew (cos t)))
   (fabs (+ ew (* (* t eh) (sin (atan (* eh (/ t ew)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -4.6e-5) || !(t <= 0.005)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs((ew + ((t * eh) * sin(atan((eh * (t / ew)))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-4.6d-5)) .or. (.not. (t <= 0.005d0))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs((ew + ((t * eh) * sin(atan((eh * (t / ew)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -4.6e-5) || !(t <= 0.005)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs((ew + ((t * eh) * Math.sin(Math.atan((eh * (t / ew)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -4.6e-5) or not (t <= 0.005):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs((ew + ((t * eh) * math.sin(math.atan((eh * (t / ew)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -4.6e-5) || !(t <= 0.005))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(ew + Float64(Float64(t * eh) * sin(atan(Float64(eh * Float64(t / ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -4.6e-5) || ~((t <= 0.005)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs((ew + ((t * eh) * sin(atan((eh * (t / ew)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -4.6e-5], N[Not[LessEqual[t, 0.005]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew + N[(N[(t * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.6e-5 or 0.0050000000000000001 < t

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 53.9%

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

      2. pow2 53.9%

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

   4. Applied egg-rr 53.9%

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

   5. Taylor expanded in ew around inf 54.0%

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

   if -4.6e-5 < t < 0.0050000000000000001

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 46.2%

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

      2. pow2 46.2%

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

   4. Applied egg-rr 46.2%

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

   5. Taylor expanded in t around 0 98.7%

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

      1. associate-*r* 98.7%

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

      2. associate-*r/ 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in t around 0 98.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-5} \lor \neg \left(t \leq 0.005\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + \left(t \cdot eh\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{t}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew * cos(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 * cos(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. add-sqr-sqrt 50.3%

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

   2. pow2 50.3%

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

4. Applied egg-rr 50.3%

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

5. Taylor expanded in ew around inf 61.0%

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

Alternative 8: 49.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -80000000 \lor \neg \left(t \leq 2.3 \cdot 10^{+57}\right):\\ \;\;\;\;ew \cdot \cos t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -80000000.0) (not (<= t 2.3e+57))) (* ew (cos t)) (fabs ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -80000000.0) || !(t <= 2.3e+57)) {
		tmp = ew * cos(t);
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-80000000.0d0)) .or. (.not. (t <= 2.3d+57))) then
        tmp = ew * cos(t)
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -80000000.0) || !(t <= 2.3e+57)) {
		tmp = ew * Math.cos(t);
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -80000000.0) or not (t <= 2.3e+57):
		tmp = ew * math.cos(t)
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -80000000.0) || !(t <= 2.3e+57))
		tmp = Float64(ew * cos(t));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -80000000.0) || ~((t <= 2.3e+57)))
		tmp = ew * cos(t);
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -80000000.0], N[Not[LessEqual[t, 2.3e+57]], $MachinePrecision]], N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], N[Abs[ew], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8e7 or 2.2999999999999999e57 < t

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 56.3%

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

      2. pow2 56.3%

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

   4. Applied egg-rr 56.3%

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

   5. Taylor expanded in ew around inf 53.7%

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

      1. *-un-lft-identity 53.7%

         \[\leadsto \color{blue}{1 \cdot \left|ew \cdot \cos t\right|} \]

      2. add-sqr-sqrt 29.5%

         \[\leadsto 1 \cdot \left|\color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}}\right| \]

      3. fabs-sqr 29.5%

         \[\leadsto 1 \cdot \color{blue}{\left(\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}\right)} \]

      4. add-sqr-sqrt 30.3%

         \[\leadsto 1 \cdot \color{blue}{\left(ew \cdot \cos t\right)} \]

      5. *-commutative 30.3%

         \[\leadsto 1 \cdot \color{blue}{\left(\cos t \cdot ew\right)} \]

   7. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{1 \cdot \left(\cos t \cdot ew\right)} \]

   if -8e7 < t < 2.2999999999999999e57

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 44.6%

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

      2. pow2 44.6%

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

   4. Applied egg-rr 44.6%

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

   5. Taylor expanded in t around 0 64.0%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -80000000 \lor \neg \left(t \leq 2.3 \cdot 10^{+57}\right):\\ \;\;\;\;ew \cdot \cos t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.2% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. add-sqr-sqrt 50.3%

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

   2. pow2 50.3%

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

4. Applied egg-rr 50.3%

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

5. Taylor expanded in t around 0 39.0%

   \[\leadsto \left|\color{blue}{ew}\right| \]
6. Add Preprocessing

Alternative 10: 43.2% accurate, 131.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-ew\\ \mathbf{else}:\\ \;\;\;\;ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (if (<= ew -1e-310) (- ew) ew))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1e-310) {
		tmp = -ew;
	} else {
		tmp = ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-1d-310)) then
        tmp = -ew
    else
        tmp = ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1e-310) {
		tmp = -ew;
	} else {
		tmp = ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1e-310:
		tmp = -ew
	else:
		tmp = ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1e-310)
		tmp = Float64(-ew);
	else
		tmp = ew;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1e-310)
		tmp = -ew;
	else
		tmp = ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1e-310], (-ew), ew]

Derivation

1. Split input into 2 regimes

2. if ew < -9.999999999999969e-311

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

   4. Applied egg-rr 26.8%

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

   5. Taylor expanded in t around 0 41.5%

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

      1. add-sqr-sqrt 41.2%

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

      2. pow2 41.2%

         \[\leadsto \color{blue}{{\left(\sqrt{\left|ew\right|}\right)}^{2}} \]

   7. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\left|ew\right|}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 41.2%

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

      2. add-sqr-sqrt 41.5%

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

      3. neg-fabs 41.5%

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

      4. add-sqr-sqrt 41.2%

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

      5. fabs-sqr 41.2%

         \[\leadsto \color{blue}{\sqrt{-ew} \cdot \sqrt{-ew}} \]

      6. add-sqr-sqrt 41.5%

         \[\leadsto \color{blue}{-ew} \]

      7. neg-sub0 41.5%

         \[\leadsto \color{blue}{0 - ew} \]

   9. Applied egg-rr 41.5%

      \[\leadsto \color{blue}{0 - ew} \]
   10. Step-by-step derivation

       1. neg-sub0 41.5%

          \[\leadsto \color{blue}{-ew} \]

   11. Simplified 41.5%

       \[\leadsto \color{blue}{-ew} \]

   if -9.999999999999969e-311 < ew

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 77.4%

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

      2. pow2 77.4%

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

   4. Applied egg-rr 77.4%

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

   5. Taylor expanded in t around 0 36.0%

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

      1. add-sqr-sqrt 35.8%

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

      2. pow2 35.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\left|ew\right|}\right)}^{2}} \]

   7. Applied egg-rr 35.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left|ew\right|}\right)}^{2}} \]

   8. Taylor expanded in ew around 0 36.0%

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

      1. rem-square-sqrt 35.8%

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

      2. fabs-sqr 35.8%

         \[\leadsto \color{blue}{\sqrt{ew} \cdot \sqrt{ew}} \]

      3. rem-square-sqrt 36.0%

         \[\leadsto \color{blue}{ew} \]

   10. Simplified 36.0%

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

Alternative 11: 21.8% accurate, 921.0× speedup

Math

\[\begin{array}{l} \\ ew \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(eh, ew, t):
	return ew

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.9%

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

   1. add-sqr-sqrt 50.3%

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

   2. pow2 50.3%

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

4. Applied egg-rr 50.3%

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

5. Taylor expanded in t around 0 39.0%

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

   1. add-sqr-sqrt 38.7%

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

   2. pow2 38.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\left|ew\right|}\right)}^{2}} \]

7. Applied egg-rr 38.7%

   \[\leadsto \color{blue}{{\left(\sqrt{\left|ew\right|}\right)}^{2}} \]

8. Taylor expanded in ew around 0 39.0%

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

   1. rem-square-sqrt 16.6%

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

   2. fabs-sqr 16.6%

      \[\leadsto \color{blue}{\sqrt{ew} \cdot \sqrt{ew}} \]

   3. rem-square-sqrt 17.7%

      \[\leadsto \color{blue}{ew} \]

10. Simplified 17.7%

    \[\leadsto \color{blue}{ew} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))