Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.1s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := \tan^{-1} \left(-\frac{eh \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew)) (t_2 (atan (- (/ (* eh (tan t)) ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -2e-259)
     (/ 1.0 (fabs (/ 1.0 ew)))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = atan(-((eh * tan(t)) / ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-259) {
		tmp = 1.0 / fabs((1.0 / 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 = cos(t) * ew
    t_2 = atan(-((eh * tan(t)) / ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-2d-259)) then
        tmp = 1.0d0 / abs((1.0d0 / 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.cos(t) * ew;
	double t_2 = Math.atan(-((eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -2e-259) {
		tmp = 1.0 / Math.abs((1.0 / ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * ew
	t_2 = math.atan(-((eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -2e-259:
		tmp = 1.0 / math.fabs((1.0 / ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = atan(Float64(-Float64(Float64(eh * tan(t)) / ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -2e-259)
		tmp = Float64(1.0 / abs(Float64(1.0 / ew)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * ew;
	t_2 = atan(-((eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-259)
		tmp = 1.0 / abs((1.0 / ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $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[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-259], N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $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))))) < -2.0000000000000001e-259

   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 73.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 40.3

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

   7. Applied rewrites 40.3%

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

   if -2.0000000000000001e-259 < (-.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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 71.1%

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

   7. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 57.6

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

   9. Applied rewrites 57.6%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(-\frac{eh \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-\frac{eh \cdot \tan t}{ew}\right) \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Cos[t], $MachinePrecision] * ew + N[(N[Sin[N[ArcTan[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-N[(eh * N[Sin[t], $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
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 4: 74.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 920000000:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -5.2e+64)
     t_1
     (if (<= eh 920000000.0) (fabs (* (cos t) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -5.2e+64) {
		tmp = t_1;
	} else if (eh <= 920000000.0) {
		tmp = fabs((cos(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (eh <= (-5.2d+64)) then
        tmp = t_1
    else if (eh <= 920000000.0d0) then
        tmp = abs((cos(t) * ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (eh <= -5.2e+64) {
		tmp = t_1;
	} else if (eh <= 920000000.0) {
		tmp = Math.abs((Math.cos(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -5.2e+64:
		tmp = t_1
	elif eh <= 920000000.0:
		tmp = math.fabs((math.cos(t) * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -5.2e+64)
		tmp = t_1;
	elseif (eh <= 920000000.0)
		tmp = abs(Float64(cos(t) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -5.2e+64)
		tmp = t_1;
	elseif (eh <= 920000000.0)
		tmp = abs((cos(t) * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -5.2e+64], t$95$1, If[LessEqual[eh, 920000000.0], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.19999999999999994e64 or 9.2e8 < 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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 53.7%

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

   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 76.1

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

   7. Applied rewrites 76.1%

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

   if -5.19999999999999994e64 < eh < 9.2e8

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   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 82.7

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

   7. Applied rewrites 82.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;eh \leq 920000000:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -4.8e-95) t_1 (if (<= t 6.5e-35) (/ 1.0 (fabs (/ 1.0 ew))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -4.8e-95) {
		tmp = t_1;
	} else if (t <= 6.5e-35) {
		tmp = 1.0 / fabs((1.0 / ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (t <= (-4.8d-95)) then
        tmp = t_1
    else if (t <= 6.5d-35) then
        tmp = 1.0d0 / abs((1.0d0 / ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (t <= -4.8e-95) {
		tmp = t_1;
	} else if (t <= 6.5e-35) {
		tmp = 1.0 / Math.abs((1.0 / ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if t <= -4.8e-95:
		tmp = t_1
	elif t <= 6.5e-35:
		tmp = 1.0 / math.fabs((1.0 / ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (t <= -4.8e-95)
		tmp = t_1;
	elseif (t <= 6.5e-35)
		tmp = Float64(1.0 / abs(Float64(1.0 / ew)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (t <= -4.8e-95)
		tmp = t_1;
	elseif (t <= 6.5e-35)
		tmp = 1.0 / abs((1.0 / ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.8e-95], t$95$1, If[LessEqual[t, 6.5e-35], N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8e-95 or 6.4999999999999999e-35 < t

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 64.6%

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

   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 61.3

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

   7. Applied rewrites 61.3%

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

   if -4.8e-95 < t < 6.4999999999999999e-35

   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 96.3%

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

   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 85.5

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

   7. Applied rewrites 85.5%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 48.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (<= t -8.5e+46) t_1 (if (<= t 7.2e-10) (/ 1.0 (fabs (/ 1.0 ew))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if (t <= -8.5e+46) {
		tmp = t_1;
	} else if (t <= 7.2e-10) {
		tmp = 1.0 / fabs((1.0 / 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 = eh * sin(t)
    if (t <= (-8.5d+46)) then
        tmp = t_1
    else if (t <= 7.2d-10) then
        tmp = 1.0d0 / abs((1.0d0 / 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 = eh * Math.sin(t);
	double tmp;
	if (t <= -8.5e+46) {
		tmp = t_1;
	} else if (t <= 7.2e-10) {
		tmp = 1.0 / Math.abs((1.0 / ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if t <= -8.5e+46:
		tmp = t_1
	elif t <= 7.2e-10:
		tmp = 1.0 / math.fabs((1.0 / ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if (t <= -8.5e+46)
		tmp = t_1;
	elseif (t <= 7.2e-10)
		tmp = Float64(1.0 / abs(Float64(1.0 / ew)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = 0.0;
	if (t <= -8.5e+46)
		tmp = t_1;
	elseif (t <= 7.2e-10)
		tmp = 1.0 / abs((1.0 / ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e+46], t$95$1, If[LessEqual[t, 7.2e-10], N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.4999999999999996e46 or 7.2e-10 < t

   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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 29.0%

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

   7. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 28.9

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

   9. Applied rewrites 28.9%

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

   if -8.4999999999999996e46 < t < 7.2e-10

   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 89.3%

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

   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 71.8

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

   7. Applied rewrites 71.8%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 45.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{if}\;ew \leq -1.65 \cdot 10^{-183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.85 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), 0.008333333333333333, eh \cdot -0.16666666666666666\right), eh\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (fabs (/ 1.0 ew)))))
   (if (<= ew -1.65e-183)
     t_1
     (if (<= ew 1.85e-142)
       (/
        1.0
        (fabs
         (/
          1.0
          (*
           t
           (fma
            (* t t)
            (fma
             (* eh (* t t))
             0.008333333333333333
             (* eh -0.16666666666666666))
            eh)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / ew));
	double tmp;
	if (ew <= -1.65e-183) {
		tmp = t_1;
	} else if (ew <= 1.85e-142) {
		tmp = 1.0 / fabs((1.0 / (t * fma((t * t), fma((eh * (t * t)), 0.008333333333333333, (eh * -0.16666666666666666)), eh))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / ew)))
	tmp = 0.0
	if (ew <= -1.65e-183)
		tmp = t_1;
	elseif (ew <= 1.85e-142)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(t * fma(Float64(t * t), fma(Float64(eh * Float64(t * t)), 0.008333333333333333, Float64(eh * -0.16666666666666666)), eh)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.65e-183], t$95$1, If[LessEqual[ew, 1.85e-142], N[(1.0 / N[Abs[N[(1.0 / 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]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.65e-183 or 1.84999999999999993e-142 < 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 81.6%

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

   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 47.9

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

   7. Applied rewrites 47.9%

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

   if -1.65e-183 < ew < 1.84999999999999993e-142

   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 43.2%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 32.9%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{-183}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{elif}\;ew \leq 1.85 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), 0.008333333333333333, eh \cdot -0.16666666666666666\right), eh\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{if}\;ew \leq -1.65 \cdot 10^{-183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.85 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t \cdot \mathsf{fma}\left(-0.16666666666666666, eh \cdot \left(t \cdot t\right), eh\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (fabs (/ 1.0 ew)))))
   (if (<= ew -1.65e-183)
     t_1
     (if (<= ew 1.85e-142)
       (/
        1.0
        (fabs (/ 1.0 (* t (fma -0.16666666666666666 (* eh (* t t)) eh)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / ew));
	double tmp;
	if (ew <= -1.65e-183) {
		tmp = t_1;
	} else if (ew <= 1.85e-142) {
		tmp = 1.0 / fabs((1.0 / (t * fma(-0.16666666666666666, (eh * (t * t)), eh))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / ew)))
	tmp = 0.0
	if (ew <= -1.65e-183)
		tmp = t_1;
	elseif (ew <= 1.85e-142)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(t * fma(-0.16666666666666666, Float64(eh * Float64(t * t)), eh)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.65e-183], t$95$1, If[LessEqual[ew, 1.85e-142], N[(1.0 / N[Abs[N[(1.0 / N[(t * N[(-0.16666666666666666 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] + eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.65e-183 or 1.84999999999999993e-142 < 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 81.6%

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

   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 47.9

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

   7. Applied rewrites 47.9%

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

   if -1.65e-183 < ew < 1.84999999999999993e-142

   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 43.2%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 32.7%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{-183}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{elif}\;ew \leq 1.85 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t \cdot \mathsf{fma}\left(-0.16666666666666666, eh \cdot \left(t \cdot t\right), eh\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.3% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{if}\;ew \leq -1.65 \cdot 10^{-183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.85 \cdot 10^{-142}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t \cdot eh}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (fabs (/ 1.0 ew)))))
   (if (<= ew -1.65e-183)
     t_1
     (if (<= ew 1.85e-142) (/ 1.0 (fabs (/ 1.0 (* t eh)))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / ew));
	double tmp;
	if (ew <= -1.65e-183) {
		tmp = t_1;
	} else if (ew <= 1.85e-142) {
		tmp = 1.0 / fabs((1.0 / (t * eh)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / abs((1.0d0 / ew))
    if (ew <= (-1.65d-183)) then
        tmp = t_1
    else if (ew <= 1.85d-142) then
        tmp = 1.0d0 / abs((1.0d0 / (t * eh)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = 1.0 / Math.abs((1.0 / ew));
	double tmp;
	if (ew <= -1.65e-183) {
		tmp = t_1;
	} else if (ew <= 1.85e-142) {
		tmp = 1.0 / Math.abs((1.0 / (t * eh)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = 1.0 / math.fabs((1.0 / ew))
	tmp = 0
	if ew <= -1.65e-183:
		tmp = t_1
	elif ew <= 1.85e-142:
		tmp = 1.0 / math.fabs((1.0 / (t * eh)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / ew)))
	tmp = 0.0
	if (ew <= -1.65e-183)
		tmp = t_1;
	elseif (ew <= 1.85e-142)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(t * eh))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = 1.0 / abs((1.0 / ew));
	tmp = 0.0;
	if (ew <= -1.65e-183)
		tmp = t_1;
	elseif (ew <= 1.85e-142)
		tmp = 1.0 / abs((1.0 / (t * eh)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.65e-183], t$95$1, If[LessEqual[ew, 1.85e-142], N[(1.0 / N[Abs[N[(1.0 / N[(t * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.65e-183 or 1.84999999999999993e-142 < 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 81.6%

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

   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 47.9

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

   7. Applied rewrites 47.9%

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

   if -1.65e-183 < ew < 1.84999999999999993e-142

   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 43.2%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 32.6%

       \[\leadsto \frac{1}{\left|\frac{1}{t \cdot \color{blue}{eh}}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 43.0% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left|\frac{1}{ew}\right|} \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (/ 1.0 (fabs (/ 1.0 ew))))

C

double code(double eh, double ew, double t) {
	return 1.0 / fabs((1.0 / 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 = 1.0d0 / abs((1.0d0 / ew))
end function

Java

public static double code(double eh, double ew, double t) {
	return 1.0 / Math.abs((1.0 / ew));
}

Python

def code(eh, ew, t):
	return 1.0 / math.fabs((1.0 / ew))

Julia

function code(eh, ew, t)
	return Float64(1.0 / abs(Float64(1.0 / ew)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = 1.0 / abs((1.0 / ew));
end

Wolfram

code[eh_, ew_, t_] := N[(1.0 / N[Abs[N[(1.0 / ew), $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 72.4%

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

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 40.4

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

7. Applied rewrites 40.4%

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
8. Add Preprocessing

Alternative 11: 20.3% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(t \cdot t, ew \cdot -0.5, ew\right) \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fma (* t t) (* ew -0.5) ew))

C

double code(double eh, double ew, double t) {
	return fma((t * t), (ew * -0.5), ew);
}

Julia

function code(eh, ew, t)
	return fma(Float64(t * t), Float64(ew * -0.5), ew)
end

Wolfram

code[eh_, ew_, t_] := N[(N[(t * t), $MachinePrecision] * N[(ew * -0.5), $MachinePrecision] + 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
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 38.0%

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

7. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft1-in N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 19.2

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

9. Applied rewrites 19.2%

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

10. Taylor expanded in eh around 0

    \[\leadsto \mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \color{blue}{ew}, ew\right) \]
11. Step-by-step derivation

12. Applied rewrites 20.6%

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

13. Final simplification 20.6%

    \[\leadsto \mathsf{fma}\left(t \cdot t, ew \cdot -0.5, ew\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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)))))))