Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.4s

Alternatives: 18

Speedup: N/A×

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, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{-ew}\\ \left|\mathsf{fma}\left(\sin t \cdot \sin \tan^{-1} t\_1, -eh, \frac{ew \cdot \cos t}{\sqrt{{t\_1}^{2} + 1}}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) (- ew)))))
   (fabs
    (fma
     (* (sin t) (sin (atan t_1)))
     (- eh)
     (/ (* ew (cos t)) (sqrt (+ (pow t_1 2.0) 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / -ew);
	return fabs(fma((sin(t) * sin(atan(t_1))), -eh, ((ew * cos(t)) / sqrt((pow(t_1, 2.0) + 1.0)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / Float64(-ew)))
	return abs(fma(Float64(sin(t) * sin(atan(t_1))), Float64(-eh), Float64(Float64(ew * cos(t)) / sqrt(Float64((t_1 ^ 2.0) + 1.0)))))
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[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-eh) + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_4 := \sin t\_3\\ \mathbf{if}\;t\_2 \cdot \cos t\_3 - t\_1 \cdot t\_4 \leq -5 \cdot 10^{-269}:\\ \;\;\;\;\left|\frac{t\_2 + eh \cdot \left(\sin t \cdot \left(eh \cdot \frac{\tan t}{ew}\right)\right)}{\sqrt{{\left(eh \cdot \frac{\tan t}{-ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\cos t}{\sqrt{{\left(\frac{eh \cdot \tan t}{ew}\right)}^{2} + 1}}, ew, t\_4 \cdot \left(-t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (* ew (cos t)))
        (t_3 (atan (/ (* (- eh) (tan t)) ew)))
        (t_4 (sin t_3)))
   (if (<= (- (* t_2 (cos t_3)) (* t_1 t_4)) -5e-269)
     (fabs
      (/
       (+ t_2 (* eh (* (sin t) (* eh (/ (tan t) ew)))))
       (sqrt (+ (pow (* eh (/ (tan t) (- ew))) 2.0) 1.0))))
     (fma
      (/ (cos t) (sqrt (+ (pow (/ (* eh (tan t)) ew) 2.0) 1.0)))
      ew
      (* t_4 (- t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = ew * cos(t);
	double t_3 = atan(((-eh * tan(t)) / ew));
	double t_4 = sin(t_3);
	double tmp;
	if (((t_2 * cos(t_3)) - (t_1 * t_4)) <= -5e-269) {
		tmp = fabs(((t_2 + (eh * (sin(t) * (eh * (tan(t) / ew))))) / sqrt((pow((eh * (tan(t) / -ew)), 2.0) + 1.0))));
	} else {
		tmp = fma((cos(t) / sqrt((pow(((eh * tan(t)) / ew), 2.0) + 1.0))), ew, (t_4 * -t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_4 = sin(t_3)
	tmp = 0.0
	if (Float64(Float64(t_2 * cos(t_3)) - Float64(t_1 * t_4)) <= -5e-269)
		tmp = abs(Float64(Float64(t_2 + Float64(eh * Float64(sin(t) * Float64(eh * Float64(tan(t) / ew))))) / sqrt(Float64((Float64(eh * Float64(tan(t) / Float64(-ew))) ^ 2.0) + 1.0))));
	else
		tmp = fma(Float64(cos(t) / sqrt(Float64((Float64(Float64(eh * tan(t)) / ew) ^ 2.0) + 1.0))), ew, Float64(t_4 * Float64(-t_1)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$3], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision], -5e-269], N[Abs[N[(N[(t$95$2 + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Cos[t], $MachinePrecision] / N[Sqrt[N[(N[Power[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(t$95$4 * (-t$95$1)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.99999999999999979e-269

   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

   4. Applied rewrites 79.7%

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

   if -4.99999999999999979e-269 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \leq -5 \cdot 10^{-269}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + eh \cdot \left(\sin t \cdot \left(eh \cdot \frac{\tan t}{ew}\right)\right)}{\sqrt{{\left(eh \cdot \frac{\tan t}{-ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\cos t}{\sqrt{{\left(\frac{eh \cdot \tan t}{ew}\right)}^{2} + 1}}, ew, \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \left(-\sin t \cdot eh\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(\sin t \cdot eh\right) \cdot \sin t\_2 \leq 10^{-106}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* (- eh) (tan t)) ew))))
   (if (<= (- (* t_1 (cos t_2)) (* (* (sin t) eh) (sin t_2))) 1e-106)
     (fabs ew)
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((sin(t) * eh) * sin(t_2))) <= 1e-106) {
		tmp = fabs(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) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((-eh * tan(t)) / ew))
    if (((t_1 * cos(t_2)) - ((sin(t) * eh) * sin(t_2))) <= 1d-106) then
        tmp = abs(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 = ew * Math.cos(t);
	double t_2 = Math.atan(((-eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((Math.sin(t) * eh) * Math.sin(t_2))) <= 1e-106) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((-eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((math.sin(t) * eh) * math.sin(t_2))) <= 1e-106:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(sin(t) * eh) * sin(t_2))) <= 1e-106)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((-eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((sin(t) * eh) * sin(t_2))) <= 1e-106)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-106], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 9.99999999999999941e-107

   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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   6. Step-by-step derivation

      1. lower-/.f64 43.5

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 43.5%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   8. Step-by-step derivation

      1. fabs-div N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|ew\right|}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|ew\right|}} \]

      3. remove-double-div N/A

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

      4. lower-fabs.f64 43.6

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

   9. Applied rewrites 43.6%

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

   if 9.99999999999999941e-107 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in eh around 0

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 64.9

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

   9. Applied rewrites 64.9%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \leq 10^{-106}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\mathsf{fma}\left(\cos t, ew, eh \cdot \left(\left(eh \cdot 0.5\right) \cdot \left(\tan t \cdot \frac{\sin t}{ew}\right)\right)\right)\right|\\ \mathbf{if}\;ew \leq -6.7 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 10^{-111}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;ew \leq 5.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(ew, ew \cdot \cos t, \left(eh \cdot eh\right) \cdot \left(\sin t \cdot \tan t\right)\right)\right|}{ew \cdot \sqrt{{\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (fma (cos t) ew (* eh (* (* eh 0.5) (* (tan t) (/ (sin t) ew))))))))
   (if (<= ew -6.7e-32)
     t_1
     (if (<= ew 1e-111)
       (fabs (* (sin t) eh))
       (if (<= ew 5.1e+75)
         (/
          (fabs (fma ew (* ew (cos t)) (* (* eh eh) (* (sin t) (tan t)))))
          (* ew (sqrt (+ (pow (* eh (/ (tan t) ew)) 2.0) 1.0))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(fma(cos(t), ew, (eh * ((eh * 0.5) * (tan(t) * (sin(t) / ew))))));
	double tmp;
	if (ew <= -6.7e-32) {
		tmp = t_1;
	} else if (ew <= 1e-111) {
		tmp = fabs((sin(t) * eh));
	} else if (ew <= 5.1e+75) {
		tmp = fabs(fma(ew, (ew * cos(t)), ((eh * eh) * (sin(t) * tan(t))))) / (ew * sqrt((pow((eh * (tan(t) / ew)), 2.0) + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(fma(cos(t), ew, Float64(eh * Float64(Float64(eh * 0.5) * Float64(tan(t) * Float64(sin(t) / ew))))))
	tmp = 0.0
	if (ew <= -6.7e-32)
		tmp = t_1;
	elseif (ew <= 1e-111)
		tmp = abs(Float64(sin(t) * eh));
	elseif (ew <= 5.1e+75)
		tmp = Float64(abs(fma(ew, Float64(ew * cos(t)), Float64(Float64(eh * eh) * Float64(sin(t) * tan(t))))) / Float64(ew * sqrt(Float64((Float64(eh * Float64(tan(t) / ew)) ^ 2.0) + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[t], $MachinePrecision] * ew + N[(eh * N[(N[(eh * 0.5), $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -6.7e-32], t$95$1, If[LessEqual[ew, 1e-111], N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 5.1e+75], N[(N[Abs[N[(ew * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * eh), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(ew * N[Sqrt[N[(N[Power[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -6.7e-32 or 5.10000000000000037e75 < 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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 70.3

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

   7. Applied rewrites 70.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh \cdot eh, 0.5 \cdot \frac{{\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \cos t\right)}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

   9. Applied rewrites 85.1%

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

   if -6.7e-32 < ew < 1.00000000000000009e-111

   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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 72.7

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

   7. Applied rewrites 72.7%

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

   if 1.00000000000000009e-111 < ew < 5.10000000000000037e75

   1. Initial program 99.6%

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in ew around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 76.0%

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

   9. Applied rewrites 76.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.7 \cdot 10^{-32}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, eh \cdot \left(\left(eh \cdot 0.5\right) \cdot \left(\tan t \cdot \frac{\sin t}{ew}\right)\right)\right)\right|\\ \mathbf{elif}\;ew \leq 10^{-111}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;ew \leq 5.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(ew, ew \cdot \cos t, \left(eh \cdot eh\right) \cdot \left(\sin t \cdot \tan t\right)\right)\right|}{ew \cdot \sqrt{{\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, eh \cdot \left(\left(eh \cdot 0.5\right) \cdot \left(\tan t \cdot \frac{\sin t}{ew}\right)\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ \mathbf{if}\;eh \leq -1.45 \cdot 10^{+149}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;eh \leq 6.2 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + eh \cdot \left(\sin t \cdot \left(eh \cdot \frac{\tan t}{ew}\right)\right)}{\sqrt{{\left(eh \cdot \frac{\tan t}{-ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_1}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)))
   (if (<= eh -1.45e+149)
     (fabs t_1)
     (if (<= eh 6.2e+112)
       (fabs
        (/
         (+ (* ew (cos t)) (* eh (* (sin t) (* eh (/ (tan t) ew)))))
         (sqrt (+ (pow (* eh (/ (tan t) (- ew))) 2.0) 1.0))))
       (/ 1.0 (fabs (/ 1.0 t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double tmp;
	if (eh <= -1.45e+149) {
		tmp = fabs(t_1);
	} else if (eh <= 6.2e+112) {
		tmp = fabs((((ew * cos(t)) + (eh * (sin(t) * (eh * (tan(t) / ew))))) / sqrt((pow((eh * (tan(t) / -ew)), 2.0) + 1.0))));
	} else {
		tmp = 1.0 / fabs((1.0 / 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 = sin(t) * eh
    if (eh <= (-1.45d+149)) then
        tmp = abs(t_1)
    else if (eh <= 6.2d+112) then
        tmp = abs((((ew * cos(t)) + (eh * (sin(t) * (eh * (tan(t) / ew))))) / sqrt((((eh * (tan(t) / -ew)) ** 2.0d0) + 1.0d0))))
    else
        tmp = 1.0d0 / abs((1.0d0 / t_1))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double tmp;
	if (eh <= -1.45e+149) {
		tmp = Math.abs(t_1);
	} else if (eh <= 6.2e+112) {
		tmp = Math.abs((((ew * Math.cos(t)) + (eh * (Math.sin(t) * (eh * (Math.tan(t) / ew))))) / Math.sqrt((Math.pow((eh * (Math.tan(t) / -ew)), 2.0) + 1.0))));
	} else {
		tmp = 1.0 / Math.abs((1.0 / t_1));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	tmp = 0
	if eh <= -1.45e+149:
		tmp = math.fabs(t_1)
	elif eh <= 6.2e+112:
		tmp = math.fabs((((ew * math.cos(t)) + (eh * (math.sin(t) * (eh * (math.tan(t) / ew))))) / math.sqrt((math.pow((eh * (math.tan(t) / -ew)), 2.0) + 1.0))))
	else:
		tmp = 1.0 / math.fabs((1.0 / t_1))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	tmp = 0.0
	if (eh <= -1.45e+149)
		tmp = abs(t_1);
	elseif (eh <= 6.2e+112)
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) + Float64(eh * Float64(sin(t) * Float64(eh * Float64(tan(t) / ew))))) / sqrt(Float64((Float64(eh * Float64(tan(t) / Float64(-ew))) ^ 2.0) + 1.0))));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	tmp = 0.0;
	if (eh <= -1.45e+149)
		tmp = abs(t_1);
	elseif (eh <= 6.2e+112)
		tmp = abs((((ew * cos(t)) + (eh * (sin(t) * (eh * (tan(t) / ew))))) / sqrt((((eh * (tan(t) / -ew)) ^ 2.0) + 1.0))));
	else
		tmp = 1.0 / abs((1.0 / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -1.45e+149], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[eh, 6.2e+112], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.4500000000000001e149

   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

   4. Applied rewrites 33.7%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 67.1

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

   7. Applied rewrites 67.1%

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

   if -1.4500000000000001e149 < eh < 6.19999999999999965e112

   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

   4. Applied rewrites 93.4%

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

   if 6.19999999999999965e112 < eh

   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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ew around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{eh \cdot \sin t}}\right|} \]

      3. lower-sin.f64 81.1

         \[\leadsto \frac{1}{\left|\frac{1}{eh \cdot \color{blue}{\sin t}}\right|} \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.45 \cdot 10^{+149}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 6.2 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + eh \cdot \left(\sin t \cdot \left(eh \cdot \frac{\tan t}{ew}\right)\right)}{\sqrt{{\left(eh \cdot \frac{\tan t}{-ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin t \cdot eh}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ \mathbf{if}\;eh \leq -1.45 \cdot 10^{+149}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;eh \leq 5.8 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + \frac{\sin t \cdot \left(\tan t \cdot \left(eh \cdot eh\right)\right)}{ew}}{\sqrt{{\left(\frac{eh \cdot \tan t}{ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_1}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)))
   (if (<= eh -1.45e+149)
     (fabs t_1)
     (if (<= eh 5.8e+112)
       (fabs
        (/
         (+ (* ew (cos t)) (/ (* (sin t) (* (tan t) (* eh eh))) ew))
         (sqrt (+ (pow (/ (* eh (tan t)) ew) 2.0) 1.0))))
       (/ 1.0 (fabs (/ 1.0 t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double tmp;
	if (eh <= -1.45e+149) {
		tmp = fabs(t_1);
	} else if (eh <= 5.8e+112) {
		tmp = fabs((((ew * cos(t)) + ((sin(t) * (tan(t) * (eh * eh))) / ew)) / sqrt((pow(((eh * tan(t)) / ew), 2.0) + 1.0))));
	} else {
		tmp = 1.0 / fabs((1.0 / 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 = sin(t) * eh
    if (eh <= (-1.45d+149)) then
        tmp = abs(t_1)
    else if (eh <= 5.8d+112) then
        tmp = abs((((ew * cos(t)) + ((sin(t) * (tan(t) * (eh * eh))) / ew)) / sqrt(((((eh * tan(t)) / ew) ** 2.0d0) + 1.0d0))))
    else
        tmp = 1.0d0 / abs((1.0d0 / t_1))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double tmp;
	if (eh <= -1.45e+149) {
		tmp = Math.abs(t_1);
	} else if (eh <= 5.8e+112) {
		tmp = Math.abs((((ew * Math.cos(t)) + ((Math.sin(t) * (Math.tan(t) * (eh * eh))) / ew)) / Math.sqrt((Math.pow(((eh * Math.tan(t)) / ew), 2.0) + 1.0))));
	} else {
		tmp = 1.0 / Math.abs((1.0 / t_1));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	tmp = 0
	if eh <= -1.45e+149:
		tmp = math.fabs(t_1)
	elif eh <= 5.8e+112:
		tmp = math.fabs((((ew * math.cos(t)) + ((math.sin(t) * (math.tan(t) * (eh * eh))) / ew)) / math.sqrt((math.pow(((eh * math.tan(t)) / ew), 2.0) + 1.0))))
	else:
		tmp = 1.0 / math.fabs((1.0 / t_1))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	tmp = 0.0
	if (eh <= -1.45e+149)
		tmp = abs(t_1);
	elseif (eh <= 5.8e+112)
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) + Float64(Float64(sin(t) * Float64(tan(t) * Float64(eh * eh))) / ew)) / sqrt(Float64((Float64(Float64(eh * tan(t)) / ew) ^ 2.0) + 1.0))));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	tmp = 0.0;
	if (eh <= -1.45e+149)
		tmp = abs(t_1);
	elseif (eh <= 5.8e+112)
		tmp = abs((((ew * cos(t)) + ((sin(t) * (tan(t) * (eh * eh))) / ew)) / sqrt(((((eh * tan(t)) / ew) ^ 2.0) + 1.0))));
	else
		tmp = 1.0 / abs((1.0 / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -1.45e+149], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[eh, 5.8e+112], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] * N[(eh * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.4500000000000001e149

   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

   4. Applied rewrites 33.7%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 67.1

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

   7. Applied rewrites 67.1%

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

   if -1.4500000000000001e149 < eh < 5.8000000000000004e112

   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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 92.9%

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

   if 5.8000000000000004e112 < eh

   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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ew around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{eh \cdot \sin t}}\right|} \]

      3. lower-sin.f64 81.1

         \[\leadsto \frac{1}{\left|\frac{1}{eh \cdot \color{blue}{\sin t}}\right|} \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.45 \cdot 10^{+149}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 5.8 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + \frac{\sin t \cdot \left(\tan t \cdot \left(eh \cdot eh\right)\right)}{ew}}{\sqrt{{\left(\frac{eh \cdot \tan t}{ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin t \cdot eh}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ \mathbf{if}\;eh \leq -6.2 \cdot 10^{+111}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\sin t, \tan t \cdot \frac{eh \cdot eh}{ew}, ew \cdot \cos t\right)\right|}{\sqrt{{\left(\frac{eh \cdot \tan t}{ew}\right)}^{2} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_1}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)))
   (if (<= eh -6.2e+111)
     (fabs t_1)
     (if (<= eh 5.5e+112)
       (/
        (fabs (fma (sin t) (* (tan t) (/ (* eh eh) ew)) (* ew (cos t))))
        (sqrt (+ (pow (/ (* eh (tan t)) ew) 2.0) 1.0)))
       (/ 1.0 (fabs (/ 1.0 t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double tmp;
	if (eh <= -6.2e+111) {
		tmp = fabs(t_1);
	} else if (eh <= 5.5e+112) {
		tmp = fabs(fma(sin(t), (tan(t) * ((eh * eh) / ew)), (ew * cos(t)))) / sqrt((pow(((eh * tan(t)) / ew), 2.0) + 1.0));
	} else {
		tmp = 1.0 / fabs((1.0 / t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	tmp = 0.0
	if (eh <= -6.2e+111)
		tmp = abs(t_1);
	elseif (eh <= 5.5e+112)
		tmp = Float64(abs(fma(sin(t), Float64(tan(t) * Float64(Float64(eh * eh) / ew)), Float64(ew * cos(t)))) / sqrt(Float64((Float64(Float64(eh * tan(t)) / ew) ^ 2.0) + 1.0)));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / t_1)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -6.2e+111], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[eh, 5.5e+112], N[(N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] * N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -6.2000000000000001e111

   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

   4. Applied rewrites 40.8%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 66.0

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

   7. Applied rewrites 66.0%

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

   if -6.2000000000000001e111 < eh < 5.50000000000000026e112

   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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 92.0%

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

   if 5.50000000000000026e112 < eh

   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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ew around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{eh \cdot \sin t}}\right|} \]

      3. lower-sin.f64 81.1

         \[\leadsto \frac{1}{\left|\frac{1}{eh \cdot \color{blue}{\sin t}}\right|} \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6.2 \cdot 10^{+111}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\sin t, \tan t \cdot \frac{eh \cdot eh}{ew}, ew \cdot \cos t\right)\right|}{\sqrt{{\left(\frac{eh \cdot \tan t}{ew}\right)}^{2} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin t \cdot eh}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+93}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;eh \leq 2.35 \cdot 10^{+112}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, eh \cdot \left(\left(eh \cdot 0.5\right) \cdot \left(\tan t \cdot \frac{\sin t}{ew}\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_1}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)))
   (if (<= eh -9e+93)
     (fabs t_1)
     (if (<= eh 2.35e+112)
       (fabs (fma (cos t) ew (* eh (* (* eh 0.5) (* (tan t) (/ (sin t) ew))))))
       (/ 1.0 (fabs (/ 1.0 t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double tmp;
	if (eh <= -9e+93) {
		tmp = fabs(t_1);
	} else if (eh <= 2.35e+112) {
		tmp = fabs(fma(cos(t), ew, (eh * ((eh * 0.5) * (tan(t) * (sin(t) / ew))))));
	} else {
		tmp = 1.0 / fabs((1.0 / t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	tmp = 0.0
	if (eh <= -9e+93)
		tmp = abs(t_1);
	elseif (eh <= 2.35e+112)
		tmp = abs(fma(cos(t), ew, Float64(eh * Float64(Float64(eh * 0.5) * Float64(tan(t) * Float64(sin(t) / ew))))));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / t_1)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -9e+93], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[eh, 2.35e+112], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew + N[(eh * N[(N[(eh * 0.5), $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -8.99999999999999981e93

   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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 65.5

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

   7. Applied rewrites 65.5%

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

   if -8.99999999999999981e93 < eh < 2.34999999999999999e112

   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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 79.9

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

   7. Applied rewrites 79.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh \cdot eh, 0.5 \cdot \frac{{\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \cos t\right)}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

   9. Applied rewrites 79.9%

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

   if 2.34999999999999999e112 < eh

   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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ew around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{eh \cdot \sin t}}\right|} \]

      3. lower-sin.f64 81.1

         \[\leadsto \frac{1}{\left|\frac{1}{eh \cdot \color{blue}{\sin t}}\right|} \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+93}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 2.35 \cdot 10^{+112}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, eh \cdot \left(\left(eh \cdot 0.5\right) \cdot \left(\tan t \cdot \frac{\sin t}{ew}\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin t \cdot eh}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+93}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;eh \leq 2.35 \cdot 10^{+112}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_1}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)))
   (if (<= eh -9e+93)
     (fabs t_1)
     (if (<= eh 2.35e+112) (fabs (* ew (cos t))) (/ 1.0 (fabs (/ 1.0 t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double tmp;
	if (eh <= -9e+93) {
		tmp = fabs(t_1);
	} else if (eh <= 2.35e+112) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = 1.0 / fabs((1.0 / 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 = sin(t) * eh
    if (eh <= (-9d+93)) then
        tmp = abs(t_1)
    else if (eh <= 2.35d+112) then
        tmp = abs((ew * cos(t)))
    else
        tmp = 1.0d0 / abs((1.0d0 / t_1))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double tmp;
	if (eh <= -9e+93) {
		tmp = Math.abs(t_1);
	} else if (eh <= 2.35e+112) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = 1.0 / Math.abs((1.0 / t_1));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	tmp = 0
	if eh <= -9e+93:
		tmp = math.fabs(t_1)
	elif eh <= 2.35e+112:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = 1.0 / math.fabs((1.0 / t_1))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	tmp = 0.0
	if (eh <= -9e+93)
		tmp = abs(t_1);
	elseif (eh <= 2.35e+112)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	tmp = 0.0;
	if (eh <= -9e+93)
		tmp = abs(t_1);
	elseif (eh <= 2.35e+112)
		tmp = abs((ew * cos(t)));
	else
		tmp = 1.0 / abs((1.0 / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -9e+93], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[eh, 2.35e+112], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -8.99999999999999981e93

   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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 65.5

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

   7. Applied rewrites 65.5%

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

   if -8.99999999999999981e93 < eh < 2.34999999999999999e112

   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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 79.7

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

   7. Applied rewrites 79.7%

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

   if 2.34999999999999999e112 < eh

   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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ew around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{eh \cdot \sin t}}\right|} \]

      3. lower-sin.f64 81.1

         \[\leadsto \frac{1}{\left|\frac{1}{eh \cdot \color{blue}{\sin t}}\right|} \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+93}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 2.35 \cdot 10^{+112}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin t \cdot eh}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot eh\right|\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.1 \cdot 10^{+112}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) eh))))
   (if (<= eh -9e+93) t_1 (if (<= eh 2.1e+112) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * eh));
	double tmp;
	if (eh <= -9e+93) {
		tmp = t_1;
	} else if (eh <= 2.1e+112) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * eh))
    if (eh <= (-9d+93)) then
        tmp = t_1
    else if (eh <= 2.1d+112) then
        tmp = abs((ew * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * eh));
	double tmp;
	if (eh <= -9e+93) {
		tmp = t_1;
	} else if (eh <= 2.1e+112) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * eh))
	tmp = 0
	if eh <= -9e+93:
		tmp = t_1
	elif eh <= 2.1e+112:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * eh))
	tmp = 0.0
	if (eh <= -9e+93)
		tmp = t_1;
	elseif (eh <= 2.1e+112)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * eh));
	tmp = 0.0;
	if (eh <= -9e+93)
		tmp = t_1;
	elseif (eh <= 2.1e+112)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -9e+93], t$95$1, If[LessEqual[eh, 2.1e+112], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.99999999999999981e93 or 2.0999999999999999e112 < eh

   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

   4. Applied rewrites 39.2%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 72.0

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

   7. Applied rewrites 72.0%

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

   if -8.99999999999999981e93 < eh < 2.0999999999999999e112

   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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 79.7

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

   7. Applied rewrites 79.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+93}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 2.1 \cdot 10^{+112}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -6.7 \cdot 10^{-32}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 8000000:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -6.7e-32)
   (fabs ew)
   (if (<= ew 8000000.0) (fabs (* (sin t) eh)) (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -6.7e-32) {
		tmp = fabs(ew);
	} else if (ew <= 8000000.0) {
		tmp = fabs((sin(t) * eh));
	} else {
		tmp = 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 (ew <= (-6.7d-32)) then
        tmp = abs(ew)
    else if (ew <= 8000000.0d0) then
        tmp = abs((sin(t) * eh))
    else
        tmp = ew * cos(t)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -6.7e-32) {
		tmp = Math.abs(ew);
	} else if (ew <= 8000000.0) {
		tmp = Math.abs((Math.sin(t) * eh));
	} else {
		tmp = ew * Math.cos(t);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -6.7e-32:
		tmp = math.fabs(ew)
	elif ew <= 8000000.0:
		tmp = math.fabs((math.sin(t) * eh))
	else:
		tmp = ew * math.cos(t)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -6.7e-32)
		tmp = abs(ew);
	elseif (ew <= 8000000.0)
		tmp = abs(Float64(sin(t) * eh));
	else
		tmp = Float64(ew * cos(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -6.7e-32)
		tmp = abs(ew);
	elseif (ew <= 8000000.0)
		tmp = abs((sin(t) * eh));
	else
		tmp = ew * cos(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -6.7e-32], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 8000000.0], N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -6.7e-32

   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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   6. Step-by-step derivation

      1. lower-/.f64 52.9

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 52.9%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   8. Step-by-step derivation

      1. fabs-div N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|ew\right|}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|ew\right|}} \]

      3. remove-double-div N/A

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

      4. lower-fabs.f64 53.0

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

   9. Applied rewrites 53.0%

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

   if -6.7e-32 < ew < 8e6

   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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 66.9

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

   7. Applied rewrites 66.9%

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

   if 8e6 < ew

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 53.8%

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

   7. Taylor expanded in eh around 0

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 66.7

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

   9. Applied rewrites 66.7%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.7 \cdot 10^{-32}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 8000000:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ \mathbf{if}\;t \leq -7 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3400:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)))
   (if (<= t -7e+29) t_1 (if (<= t 3400.0) (fabs ew) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double tmp;
	if (t <= -7e+29) {
		tmp = t_1;
	} else if (t <= 3400.0) {
		tmp = fabs(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 = sin(t) * eh
    if (t <= (-7d+29)) then
        tmp = t_1
    else if (t <= 3400.0d0) then
        tmp = abs(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.sin(t) * eh;
	double tmp;
	if (t <= -7e+29) {
		tmp = t_1;
	} else if (t <= 3400.0) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	tmp = 0
	if t <= -7e+29:
		tmp = t_1
	elif t <= 3400.0:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	tmp = 0.0
	if (t <= -7e+29)
		tmp = t_1;
	elseif (t <= 3400.0)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	tmp = 0.0;
	if (t <= -7e+29)
		tmp = t_1;
	elseif (t <= 3400.0)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[t, -7e+29], t$95$1, If[LessEqual[t, 3400.0], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.99999999999999958e29 or 3400 < t

   1. Initial program 99.5%

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 33.1%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \color{blue}{eh \cdot \sin t} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{eh \cdot \sin t} \]

      2. lower-sin.f64 27.9

         \[\leadsto eh \cdot \color{blue}{\sin t} \]

   9. Applied rewrites 27.9%

      \[\leadsto \color{blue}{eh \cdot \sin t} \]

   if -6.99999999999999958e29 < t < 3400

   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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   6. Step-by-step derivation

      1. lower-/.f64 66.2

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 66.2%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   8. Step-by-step derivation

      1. fabs-div N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|ew\right|}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|ew\right|}} \]

      3. remove-double-div N/A

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

      4. lower-fabs.f64 66.4

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

   9. Applied rewrites 66.4%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+29}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{elif}\;t \leq 3400:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\sin t \cdot eh\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.8% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), -0.008333333333333333, eh \cdot 0.16666666666666666\right), -eh\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 5.2e+112)
   (fabs ew)
   (*
    t
    (fma
     (* t t)
     (fma (* eh (* t t)) -0.008333333333333333 (* eh 0.16666666666666666))
     (- eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 5.2e+112) {
		tmp = fabs(ew);
	} else {
		tmp = t * fma((t * t), fma((eh * (t * t)), -0.008333333333333333, (eh * 0.16666666666666666)), -eh);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 5.2e+112)
		tmp = abs(ew);
	else
		tmp = Float64(t * fma(Float64(t * t), fma(Float64(eh * Float64(t * t)), -0.008333333333333333, Float64(eh * 0.16666666666666666)), Float64(-eh)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 5.2e+112], N[Abs[ew], $MachinePrecision], N[(t * N[(N[(t * t), $MachinePrecision] * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] * -0.008333333333333333 + N[(eh * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-eh)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 5.2000000000000001e112

   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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   6. Step-by-step derivation

      1. lower-/.f64 46.4

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 46.4%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   8. Step-by-step derivation

      1. fabs-div N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|ew\right|}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|ew\right|}} \]

      3. remove-double-div N/A

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

      4. lower-fabs.f64 46.5

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

   9. Applied rewrites 46.5%

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

   if 5.2000000000000001e112 < eh

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 3.8%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sin t\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(eh \cdot \sin t\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin t \cdot eh}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin t \cdot \left(\mathsf{neg}\left(eh\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin t \cdot \color{blue}{\left(-1 \cdot eh\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin t \cdot \left(-1 \cdot eh\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin t} \cdot \left(-1 \cdot eh\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sin t \cdot \color{blue}{\left(\mathsf{neg}\left(eh\right)\right)} \]

      8. lower-neg.f64 48.3

         \[\leadsto \sin t \cdot \color{blue}{\left(-eh\right)} \]

   9. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\sin t \cdot \left(-eh\right)} \]

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{t \cdot \left(-1 \cdot eh + {t}^{2} \cdot \left(\frac{-1}{120} \cdot \left(eh \cdot {t}^{2}\right) + \frac{1}{6} \cdot eh\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto t \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{-1}{120} \cdot \left(eh \cdot {t}^{2}\right) + \frac{1}{6} \cdot eh\right) + -1 \cdot eh\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto t \cdot \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{-1}{120} \cdot \left(eh \cdot {t}^{2}\right) + \frac{1}{6} \cdot eh, -1 \cdot eh\right)} \]

       4. unpow2 N/A

          \[\leadsto t \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{-1}{120} \cdot \left(eh \cdot {t}^{2}\right) + \frac{1}{6} \cdot eh, -1 \cdot eh\right) \]

       5. lower-*.f64 N/A

          \[\leadsto t \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{-1}{120} \cdot \left(eh \cdot {t}^{2}\right) + \frac{1}{6} \cdot eh, -1 \cdot eh\right) \]

       6. *-commutative N/A

          \[\leadsto t \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{\left(eh \cdot {t}^{2}\right) \cdot \frac{-1}{120}} + \frac{1}{6} \cdot eh, -1 \cdot eh\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto t \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left(eh \cdot {t}^{2}, \frac{-1}{120}, \frac{1}{6} \cdot eh\right)}, -1 \cdot eh\right) \]

       8. lower-*.f64 N/A

          \[\leadsto t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{eh \cdot {t}^{2}}, \frac{-1}{120}, \frac{1}{6} \cdot eh\right), -1 \cdot eh\right) \]

       9. unpow2 N/A

          \[\leadsto t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \color{blue}{\left(t \cdot t\right)}, \frac{-1}{120}, \frac{1}{6} \cdot eh\right), -1 \cdot eh\right) \]

       10. lower-*.f64 N/A

           \[\leadsto t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \color{blue}{\left(t \cdot t\right)}, \frac{-1}{120}, \frac{1}{6} \cdot eh\right), -1 \cdot eh\right) \]

       11. *-commutative N/A

           \[\leadsto t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), \frac{-1}{120}, \color{blue}{eh \cdot \frac{1}{6}}\right), -1 \cdot eh\right) \]

       12. lower-*.f64 N/A

           \[\leadsto t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), \frac{-1}{120}, \color{blue}{eh \cdot \frac{1}{6}}\right), -1 \cdot eh\right) \]

       13. mul-1-neg N/A

           \[\leadsto t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), \frac{-1}{120}, eh \cdot \frac{1}{6}\right), \color{blue}{\mathsf{neg}\left(eh\right)}\right) \]

       14. lower-neg.f64 30.4

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

   12. Applied rewrites 30.4%

       \[\leadsto \color{blue}{t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), -0.008333333333333333, eh \cdot 0.16666666666666666\right), -eh\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 41.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \left(\left(-t\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 5.2e+112)
   (fabs ew)
   (*
    eh
    (*
     (- t)
     (fma
      (* t t)
      (fma (* t t) 0.008333333333333333 -0.16666666666666666)
      1.0)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 5.2e+112) {
		tmp = fabs(ew);
	} else {
		tmp = eh * (-t * fma((t * t), fma((t * t), 0.008333333333333333, -0.16666666666666666), 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 5.2e+112)
		tmp = abs(ew);
	else
		tmp = Float64(eh * Float64(Float64(-t) * fma(Float64(t * t), fma(Float64(t * t), 0.008333333333333333, -0.16666666666666666), 1.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 5.2e+112], N[Abs[ew], $MachinePrecision], N[(eh * N[((-t) * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 5.2000000000000001e112

   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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   6. Step-by-step derivation

      1. lower-/.f64 46.4

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 46.4%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   8. Step-by-step derivation

      1. fabs-div N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|ew\right|}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|ew\right|}} \]

      3. remove-double-div N/A

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

      4. lower-fabs.f64 46.5

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

   9. Applied rewrites 46.5%

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

   if 5.2000000000000001e112 < eh

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 3.8%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sin t\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(eh \cdot \sin t\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin t \cdot eh}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin t \cdot \left(\mathsf{neg}\left(eh\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin t \cdot \color{blue}{\left(-1 \cdot eh\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin t \cdot \left(-1 \cdot eh\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin t} \cdot \left(-1 \cdot eh\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sin t \cdot \color{blue}{\left(\mathsf{neg}\left(eh\right)\right)} \]

      8. lower-neg.f64 48.3

         \[\leadsto \sin t \cdot \color{blue}{\left(-eh\right)} \]

   9. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\sin t \cdot \left(-eh\right)} \]

   10. Taylor expanded in t around 0

       \[\leadsto \color{blue}{\left(t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \left(\mathsf{neg}\left(eh\right)\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \left(t \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(t \cdot \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{120} \cdot {t}^{2} - \frac{1}{6}, 1\right)}\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \left(t \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{120} \cdot {t}^{2} - \frac{1}{6}, 1\right)\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(t \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{120} \cdot {t}^{2} - \frac{1}{6}, 1\right)\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \left(t \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{\frac{1}{120} \cdot {t}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(t \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(t \cdot \mathsf{fma}\left(t \cdot t, {t}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, 1\right)\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto \left(t \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \left(\mathsf{neg}\left(eh\right)\right) \]

       11. lower-*.f64 30.3

           \[\leadsto \left(t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \left(-eh\right) \]

   12. Applied rewrites 30.3%

       \[\leadsto \color{blue}{\left(t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \cdot \left(-eh\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \left(\left(-t\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.9% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(eh \cdot \left(t \cdot t\right)\right) \cdot 0.16666666666666666 - eh\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 5.2e+112)
   (fabs ew)
   (* t (- (* (* eh (* t t)) 0.16666666666666666) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 5.2e+112) {
		tmp = fabs(ew);
	} else {
		tmp = t * (((eh * (t * t)) * 0.16666666666666666) - 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 <= 5.2d+112) then
        tmp = abs(ew)
    else
        tmp = t * (((eh * (t * t)) * 0.16666666666666666d0) - eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 5.2e+112) {
		tmp = Math.abs(ew);
	} else {
		tmp = t * (((eh * (t * t)) * 0.16666666666666666) - eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 5.2e+112:
		tmp = math.fabs(ew)
	else:
		tmp = t * (((eh * (t * t)) * 0.16666666666666666) - eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 5.2e+112)
		tmp = abs(ew);
	else
		tmp = Float64(t * Float64(Float64(Float64(eh * Float64(t * t)) * 0.16666666666666666) - eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 5.2e+112)
		tmp = abs(ew);
	else
		tmp = t * (((eh * (t * t)) * 0.16666666666666666) - eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 5.2e+112], N[Abs[ew], $MachinePrecision], N[(t * N[(N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - eh), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 5.2000000000000001e112

   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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   6. Step-by-step derivation

      1. lower-/.f64 46.4

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 46.4%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   8. Step-by-step derivation

      1. fabs-div N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|ew\right|}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|ew\right|}} \]

      3. remove-double-div N/A

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

      4. lower-fabs.f64 46.5

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

   9. Applied rewrites 46.5%

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

   if 5.2000000000000001e112 < eh

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 3.8%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sin t\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(eh \cdot \sin t\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin t \cdot eh}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin t \cdot \left(\mathsf{neg}\left(eh\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin t \cdot \color{blue}{\left(-1 \cdot eh\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin t \cdot \left(-1 \cdot eh\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin t} \cdot \left(-1 \cdot eh\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sin t \cdot \color{blue}{\left(\mathsf{neg}\left(eh\right)\right)} \]

      8. lower-neg.f64 48.3

         \[\leadsto \sin t \cdot \color{blue}{\left(-eh\right)} \]

   9. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\sin t \cdot \left(-eh\right)} \]

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

          \[\leadsto t \cdot \left(\frac{1}{6} \cdot \left(eh \cdot \color{blue}{\left(t \cdot t\right)}\right) - eh\right) \]

       9. lower-*.f64 29.8

          \[\leadsto t \cdot \left(0.16666666666666666 \cdot \left(eh \cdot \color{blue}{\left(t \cdot t\right)}\right) - eh\right) \]

   12. Applied rewrites 29.8%

       \[\leadsto \color{blue}{t \cdot \left(0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right) - eh\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(eh \cdot \left(t \cdot t\right)\right) \cdot 0.16666666666666666 - eh\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.9% accurate, 61.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 1.28 \cdot 10^{+113}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-eh\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 1.28e+113) (fabs ew) (* t (- eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.28e+113) {
		tmp = fabs(ew);
	} else {
		tmp = t * -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 <= 1.28d+113) then
        tmp = abs(ew)
    else
        tmp = t * -eh
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.28e+113) {
		tmp = Math.abs(ew);
	} else {
		tmp = t * -eh;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 1.28e+113:
		tmp = math.fabs(ew)
	else:
		tmp = t * -eh
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 1.28e+113)
		tmp = abs(ew);
	else
		tmp = Float64(t * Float64(-eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 1.28e+113)
		tmp = abs(ew);
	else
		tmp = t * -eh;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 1.28e+113], N[Abs[ew], $MachinePrecision], N[(t * (-eh)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.27999999999999994e113

   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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   6. Step-by-step derivation

      1. lower-/.f64 46.4

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 46.4%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   8. Step-by-step derivation

      1. fabs-div N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|ew\right|}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|ew\right|}} \]

      3. remove-double-div N/A

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

      4. lower-fabs.f64 46.5

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

   9. Applied rewrites 46.5%

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

   if 1.27999999999999994e113 < eh

   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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 3.8%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sin t\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(eh \cdot \sin t\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin t \cdot eh}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin t \cdot \left(\mathsf{neg}\left(eh\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin t \cdot \color{blue}{\left(-1 \cdot eh\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin t \cdot \left(-1 \cdot eh\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin t} \cdot \left(-1 \cdot eh\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sin t \cdot \color{blue}{\left(\mathsf{neg}\left(eh\right)\right)} \]

      8. lower-neg.f64 48.3

         \[\leadsto \sin t \cdot \color{blue}{\left(-eh\right)} \]

   9. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\sin t \cdot \left(-eh\right)} \]

   10. Taylor expanded in t around 0

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

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(eh \cdot t\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{t \cdot eh}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(eh\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(-1 \cdot eh\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{t \cdot \left(-1 \cdot eh\right)} \]

       6. mul-1-neg N/A

          \[\leadsto t \cdot \color{blue}{\left(\mathsf{neg}\left(eh\right)\right)} \]

       7. lower-neg.f64 29.3

          \[\leadsto t \cdot \color{blue}{\left(-eh\right)} \]

   12. Applied rewrites 29.3%

       \[\leadsto \color{blue}{t \cdot \left(-eh\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 43.5% accurate, 287.3× 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 99.8%

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

4. Applied rewrites 77.6%

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

5. Taylor expanded in t around 0

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
6. Step-by-step derivation

   1. lower-/.f64 41.6

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

7. Applied rewrites 41.6%

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
8. Step-by-step derivation

   1. fabs-div N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|ew\right|}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|ew\right|}} \]

   3. remove-double-div N/A

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

   4. lower-fabs.f64 41.7

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

9. Applied rewrites 41.7%

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

Alternative 18: 22.3% accurate, 862.0× 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 99.8%

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

4. Applied rewrites 77.6%

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

5. Taylor expanded in t around 0

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
6. Step-by-step derivation

   1. lower-/.f64 41.6

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

7. Applied rewrites 41.6%

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
8. Step-by-step derivation

   1. inv-pow N/A

      \[\leadsto \frac{1}{\left|\color{blue}{{ew}^{-1}}\right|} \]

   2. sqr-pow N/A

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

   3. fabs-sqr N/A

      \[\leadsto \frac{1}{\color{blue}{{ew}^{\left(\frac{-1}{2}\right)} \cdot {ew}^{\left(\frac{-1}{2}\right)}}} \]

   4. sqr-pow N/A

      \[\leadsto \frac{1}{\color{blue}{{ew}^{-1}}} \]

   5. pow-flip N/A

      \[\leadsto \color{blue}{{ew}^{\left(\mathsf{neg}\left(-1\right)\right)}} \]

   6. metadata-eval N/A

      \[\leadsto {ew}^{\color{blue}{1}} \]

   7. unpow1 20.2

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

9. Applied rewrites 20.2%

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

Reproduce

herbie shell --seed 2024214 
(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)))))))