Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.2s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * tan(t))
	return abs(fma(Float64(cos(t) / sqrt(Float64(1.0 + (Float64(t_1 / ew) ^ 2.0)))), ew, Float64(eh * Float64(sin(atan(Float64(t_1 / Float64(-ew)))) * Float64(-sin(t))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Cos[t], $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(t$95$1 / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(eh * N[(N[Sin[N[ArcTan[N[(t$95$1 / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-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. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-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 egg-rr 99.8%

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

   1. /-lowering-/.f64 N/A

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

   2. cos-lowering-cos.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. +-lowering-+.f64 N/A

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

   7. unpow2 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. distribute-frac-neg2 N/A

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

   10. sqr-neg N/A

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

   11. pow2 N/A

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

   12. pow-lowering-pow.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. tan-lowering-tan.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-254}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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-254)
     (fabs 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-254) {
		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 = cos(t) * ew
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-2d-254)) 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.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-254) {
		tmp = Math.abs(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-254:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -2e-254)
		tmp = abs(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-254)
		tmp = abs(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-254], 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))))) < -1.9999999999999998e-254

   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. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-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 egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 39.0%

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

   if -1.9999999999999998e-254 < (-.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 99.9%

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

   4. Applied egg-rr 72.8%

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

   5. Taylor expanded in ew around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. cos-lowering-cos.f64 63.2

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

   7. Simplified 63.2%

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

      1. inv-pow N/A

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

      2. *-commutative N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\cos t \cdot ew}}} \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{\cos t \cdot ew}{1}} \]

      8. /-rgt-identity N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cos-lowering-cos.f64 63.3

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

   9. Applied egg-rr 63.3%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-254}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-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 egg-rr 99.8%

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

   1. /-lowering-/.f64 N/A

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

   2. cos-lowering-cos.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. +-lowering-+.f64 N/A

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

   7. unpow2 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. distribute-frac-neg2 N/A

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

   10. sqr-neg N/A

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

   11. pow2 N/A

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

   12. pow-lowering-pow.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. tan-lowering-tan.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in eh around 0

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

   1. mul-1-neg N/A

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

   2. associate-*r* N/A

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

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

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

   4. mul-1-neg N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 4: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Cos[t], $MachinePrecision] * ew + N[(eh * N[(N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-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. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-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 egg-rr 99.8%

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

5. Taylor expanded in eh around 0

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

   1. cos-lowering-cos.f64 99.1

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 75.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos t \cdot ew\right|\\ \mathbf{if}\;ew \leq -3.8 \cdot 10^{-100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (cos t) ew))))
   (if (<= ew -3.8e-100) t_1 (if (<= ew 4.2e-21) (fabs (* eh (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(t) * ew));
	double tmp;
	if (ew <= -3.8e-100) {
		tmp = t_1;
	} else if (ew <= 4.2e-21) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((cos(t) * ew))
    if (ew <= (-3.8d-100)) then
        tmp = t_1
    else if (ew <= 4.2d-21) then
        tmp = abs((eh * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.cos(t) * ew));
	double tmp;
	if (ew <= -3.8e-100) {
		tmp = t_1;
	} else if (ew <= 4.2e-21) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(t) * ew))
	tmp = 0
	if ew <= -3.8e-100:
		tmp = t_1
	elif ew <= 4.2e-21:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(t) * ew))
	tmp = 0.0
	if (ew <= -3.8e-100)
		tmp = t_1;
	elseif (ew <= 4.2e-21)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(t) * ew));
	tmp = 0.0;
	if (ew <= -3.8e-100)
		tmp = t_1;
	elseif (ew <= 4.2e-21)
		tmp = abs((eh * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3.8e-100], t$95$1, If[LessEqual[ew, 4.2e-21], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.79999999999999997e-100 or 4.20000000000000025e-21 < ew

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-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 egg-rr 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. *-lowering-*.f64 N/A

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

      2. cos-lowering-cos.f64 86.3

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

   7. Simplified 86.3%

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

   if -3.79999999999999997e-100 < ew < 4.20000000000000025e-21

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. sin-atan N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

   4. Applied egg-rr 41.5%

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

   5. Taylor expanded in eh around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 74.7

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

   7. Simplified 74.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.8 \cdot 10^{-100}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;ew \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -4.5 \cdot 10^{-100}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 5.7 \cdot 10^{-21}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -4.5e-100)
   (fabs ew)
   (if (<= ew 5.7e-21) (fabs (* eh (sin t))) (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.5e-100) {
		tmp = fabs(ew);
	} else if (ew <= 5.7e-21) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-4.5d-100)) then
        tmp = abs(ew)
    else if (ew <= 5.7d-21) then
        tmp = abs((eh * sin(t)))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.5e-100) {
		tmp = Math.abs(ew);
	} else if (ew <= 5.7e-21) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -4.5e-100:
		tmp = math.fabs(ew)
	elif ew <= 5.7e-21:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -4.5e-100)
		tmp = abs(ew);
	elseif (ew <= 5.7e-21)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -4.5e-100)
		tmp = abs(ew);
	elseif (ew <= 5.7e-21)
		tmp = abs((eh * sin(t)));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -4.5e-100], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 5.7e-21], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.5000000000000001e-100

   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. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-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 egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 49.1%

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

   if -4.5000000000000001e-100 < ew < 5.6999999999999996e-21

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. sin-atan N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

   4. Applied egg-rr 41.5%

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

   5. Taylor expanded in eh around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 74.7

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

   7. Simplified 74.7%

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

   if 5.6999999999999996e-21 < ew

   1. Initial program 99.9%

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

   4. Applied egg-rr 92.8%

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

   5. Taylor expanded in ew around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. cos-lowering-cos.f64 89.5

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

   7. Simplified 89.5%

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

      1. inv-pow N/A

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

      2. *-commutative N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

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

      6. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\cos t \cdot ew}}} \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{\cos t \cdot ew}{1}} \]

      8. /-rgt-identity N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cos-lowering-cos.f64 61.9

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

   9. Applied egg-rr 61.9%

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

Alternative 7: 46.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;eh \leq -1.05 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.3 \cdot 10^{+180}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (<= eh -1.05e+98) t_1 (if (<= eh 6.3e+180) (fabs ew) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if (eh <= -1.05e+98) {
		tmp = t_1;
	} else if (eh <= 6.3e+180) {
		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 = eh * sin(t)
    if (eh <= (-1.05d+98)) then
        tmp = t_1
    else if (eh <= 6.3d+180) 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 = eh * Math.sin(t);
	double tmp;
	if (eh <= -1.05e+98) {
		tmp = t_1;
	} else if (eh <= 6.3e+180) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if eh <= -1.05e+98:
		tmp = t_1
	elif eh <= 6.3e+180:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if (eh <= -1.05e+98)
		tmp = t_1;
	elseif (eh <= 6.3e+180)
		tmp = abs(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 (eh <= -1.05e+98)
		tmp = t_1;
	elseif (eh <= 6.3e+180)
		tmp = abs(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[eh, -1.05e+98], t$95$1, If[LessEqual[eh, 6.3e+180], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.05000000000000002e98 or 6.2999999999999999e180 < 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 egg-rr 40.6%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. fabs-sqr N/A

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

      4. sqr-pow N/A

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

   6. Applied egg-rr 18.2%

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

   7. Taylor expanded in eh around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 49.2

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

   9. Simplified 49.2%

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

   if -1.05000000000000002e98 < eh < 6.2999999999999999e180

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-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 egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 47.9%

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

Alternative 8: 45.6% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -4.9 \cdot 10^{-114}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 1.6 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t \cdot eh}\right|}\\ \mathbf{else}:\\ \;\;\;\;ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -4.9e-114)
   (fabs ew)
   (if (<= ew 1.6e-218) (/ 1.0 (fabs (/ 1.0 (* t eh)))) ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.9e-114) {
		tmp = fabs(ew);
	} else if (ew <= 1.6e-218) {
		tmp = 1.0 / fabs((1.0 / (t * eh)));
	} else {
		tmp = ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-4.9d-114)) then
        tmp = abs(ew)
    else if (ew <= 1.6d-218) then
        tmp = 1.0d0 / abs((1.0d0 / (t * eh)))
    else
        tmp = ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.9e-114) {
		tmp = Math.abs(ew);
	} else if (ew <= 1.6e-218) {
		tmp = 1.0 / Math.abs((1.0 / (t * eh)));
	} else {
		tmp = ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -4.9e-114:
		tmp = math.fabs(ew)
	elif ew <= 1.6e-218:
		tmp = 1.0 / math.fabs((1.0 / (t * eh)))
	else:
		tmp = ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -4.9e-114)
		tmp = abs(ew);
	elseif (ew <= 1.6e-218)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(t * eh))));
	else
		tmp = ew;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -4.9e-114)
		tmp = abs(ew);
	elseif (ew <= 1.6e-218)
		tmp = 1.0 / abs((1.0 / (t * eh)));
	else
		tmp = ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -4.9e-114], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 1.6e-218], N[(1.0 / N[Abs[N[(1.0 / N[(t * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], ew]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.8999999999999997e-114

   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. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-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 egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 48.7%

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

   if -4.8999999999999997e-114 < ew < 1.6000000000000001e-218

   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 egg-rr 35.1%

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

   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. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 47.2

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

   10. Simplified 47.2%

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

   if 1.6000000000000001e-218 < 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 egg-rr 85.0%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. fabs-sqr N/A

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

      4. sqr-pow N/A

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

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in t around 0

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

   9. Simplified 47.3%

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

Alternative 9: 42.5% 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
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-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 egg-rr 99.8%

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

5. Taylor expanded in t around 0

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

7. Simplified 40.6%

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

Alternative 10: 22.0% 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 egg-rr 75.4%

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

   1. inv-pow N/A

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

   2. sqr-pow N/A

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

   3. fabs-sqr N/A

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

   4. sqr-pow N/A

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

6. Applied egg-rr 37.2%

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

7. Taylor expanded in t around 0

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

9. Simplified 22.9%

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

Reproduce

herbie shell --seed 2024205 
(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)))))))