Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.3s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[((-N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]) * N[Cos[t], $MachinePrecision] + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 75.4%

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

5. Taylor expanded in eh around -inf

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

   1. lower-cos.f64 98.3

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 3: 98.0% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (fma (- (* eh (tan t))) (cos t) (/ (* ew (cos t)) 1.0))))

C

double code(double eh, double ew, double t) {
	return fabs(fma(-(eh * tan(t)), cos(t), ((ew * cos(t)) / 1.0)));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(-Float64(eh * tan(t))), cos(t), Float64(Float64(ew * cos(t)) / 1.0)))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[((-N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]) * N[Cos[t], $MachinePrecision] + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / 1.0), $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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 75.4%

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

5. Taylor expanded in eh around -inf

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

   1. lower-cos.f64 98.3

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 97.9%

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

11. Final simplification 97.9%

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

Alternative 4: 74.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+24}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+73}:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh \cdot \tan t, \cos t, \frac{\mathsf{fma}\left(-0.5, ew \cdot \left(t \cdot t\right), ew\right)}{1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -8e+24)
   (fabs (* ew (cos t)))
   (if (<= t 7.2e+73)
     (fabs
      (fma (- (* eh (tan t))) (cos t) (/ (fma -0.5 (* ew (* t t)) ew) 1.0)))
     (fabs (* eh (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -8e+24) {
		tmp = fabs((ew * cos(t)));
	} else if (t <= 7.2e+73) {
		tmp = fabs(fma(-(eh * tan(t)), cos(t), (fma(-0.5, (ew * (t * t)), ew) / 1.0)));
	} else {
		tmp = fabs((eh * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -8e+24)
		tmp = abs(Float64(ew * cos(t)));
	elseif (t <= 7.2e+73)
		tmp = abs(fma(Float64(-Float64(eh * tan(t))), cos(t), Float64(fma(-0.5, Float64(ew * Float64(t * t)), ew) / 1.0)));
	else
		tmp = abs(Float64(eh * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -8e+24], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 7.2e+73], N[Abs[N[((-N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]) * N[Cos[t], $MachinePrecision] + N[(N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.9999999999999999e24

   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

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 56.3

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

   7. Applied rewrites 56.3%

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

   if -7.9999999999999999e24 < t < 7.1999999999999998e73

   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. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in eh around -inf

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

      1. lower-cos.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in eh around 0

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

   10. Applied rewrites 98.3%

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

   11. Taylor expanded in t around 0

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 93.9

          \[\leadsto \left|\mathsf{fma}\left(\left(-eh\right) \cdot \tan t, \cos t, \frac{\mathsf{fma}\left(-0.5, ew \cdot \color{blue}{\left(t \cdot t\right)}, ew\right)}{1}\right)\right| \]

   13. Applied rewrites 93.9%

       \[\leadsto \left|\mathsf{fma}\left(\left(-eh\right) \cdot \tan t, \cos t, \frac{\color{blue}{\mathsf{fma}\left(-0.5, ew \cdot \left(t \cdot t\right), ew\right)}}{1}\right)\right| \]

   if 7.1999999999999998e73 < t

   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

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 67.0

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

   7. Applied rewrites 67.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+24}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+73}:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh \cdot \tan t, \cos t, \frac{\mathsf{fma}\left(-0.5, ew \cdot \left(t \cdot t\right), ew\right)}{1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.075:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.00035:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh, t \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{-ew}\right), ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -0.075)
   (fabs (* ew (cos t)))
   (if (<= t 0.00035)
     (fabs (fma (- eh) (* t (sin (atan (/ (* eh t) (- ew))))) ew))
     (fabs (* eh (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -0.075) {
		tmp = fabs((ew * cos(t)));
	} else if (t <= 0.00035) {
		tmp = fabs(fma(-eh, (t * sin(atan(((eh * t) / -ew)))), ew));
	} else {
		tmp = fabs((eh * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -0.075)
		tmp = abs(Float64(ew * cos(t)));
	elseif (t <= 0.00035)
		tmp = abs(fma(Float64(-eh), Float64(t * sin(atan(Float64(Float64(eh * t) / Float64(-ew))))), ew));
	else
		tmp = abs(Float64(eh * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -0.075], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 0.00035], N[Abs[N[((-eh) * N[(t * N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.0749999999999999972

   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

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 55.2

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

   7. Applied rewrites 55.2%

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

   if -0.0749999999999999972 < t < 3.49999999999999996e-4

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-tan.f64 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 98.0%

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

   if 3.49999999999999996e-4 < t

   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

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 69.4

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

   7. Applied rewrites 69.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.075:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.00035:\\ \;\;\;\;\left|\mathsf{fma}\left(-eh, t \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{-ew}\right), ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -3.9 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.2 \cdot 10^{-41}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -3.9e-74) t_1 (if (<= ew 4.2e-41) (fabs (* eh (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -3.9e-74) {
		tmp = t_1;
	} else if (ew <= 4.2e-41) {
		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((ew * cos(t)))
    if (ew <= (-3.9d-74)) then
        tmp = t_1
    else if (ew <= 4.2d-41) 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((ew * Math.cos(t)));
	double tmp;
	if (ew <= -3.9e-74) {
		tmp = t_1;
	} else if (ew <= 4.2e-41) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -3.9e-74:
		tmp = t_1
	elif ew <= 4.2e-41:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -3.9e-74)
		tmp = t_1;
	elseif (ew <= 4.2e-41)
		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((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -3.9e-74)
		tmp = t_1;
	elseif (ew <= 4.2e-41)
		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[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3.9e-74], t$95$1, If[LessEqual[ew, 4.2e-41], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.9000000000000001e-74 or 4.20000000000000025e-41 < 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. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if -3.9000000000000001e-74 < ew < 4.20000000000000025e-41

   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

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 77.0

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

   7. Applied rewrites 77.0%

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

Alternative 7: 60.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-118}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh \cdot \left(eh \cdot \frac{t \cdot t}{ew}\right), 0.5, \mathsf{fma}\left(-0.5, ew \cdot \left(t \cdot t\right), ew\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -1.02e-47)
     t_1
     (if (<= t 8.2e-118)
       (fabs
        (fma (* eh (* eh (/ (* t t) ew))) 0.5 (fma -0.5 (* ew (* t t)) ew)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -1.02e-47) {
		tmp = t_1;
	} else if (t <= 8.2e-118) {
		tmp = fabs(fma((eh * (eh * ((t * t) / ew))), 0.5, fma(-0.5, (ew * (t * 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 (t <= -1.02e-47)
		tmp = t_1;
	elseif (t <= 8.2e-118)
		tmp = abs(fma(Float64(eh * Float64(eh * Float64(Float64(t * t) / ew))), 0.5, fma(-0.5, Float64(ew * Float64(t * t)), ew)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.02e-47], t$95$1, If[LessEqual[t, 8.2e-118], N[Abs[N[(N[(eh * N[(eh * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.02000000000000002e-47 or 8.2000000000000006e-118 < 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

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if -1.02000000000000002e-47 < t < 8.2000000000000006e-118

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in eh around 0

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

   10. Applied rewrites 68.8%

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

   12. Applied rewrites 78.3%

       \[\leadsto \left|\mathsf{fma}\left(eh \cdot \left(eh \cdot \frac{t \cdot t}{ew}\right), 0.5, \mathsf{fma}\left(-0.5, ew \cdot \left(t \cdot t\right), ew\right)\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 39.1% accurate, 45.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision] + ew), $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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 75.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. distribute-lft1-in N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 31.2

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

7. Applied rewrites 31.2%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 35.9%

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

Alternative 9: 4.9% accurate, 47.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((-0.5d0) * (ew * (t * t))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((-0.5 * (ew * (t * t))));
}

Python

def code(eh, ew, t):
	return math.fabs((-0.5 * (ew * (t * t))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((-0.5 * (ew * (t * t))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(-0.5 * N[(ew * N[(t * t), $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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 75.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. distribute-lft1-in N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 31.2

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

7. Applied rewrites 31.2%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 35.9%

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

11. Taylor expanded in t around inf

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

13. Applied rewrites 4.8%

    \[\leadsto \left|-0.5 \cdot \left(ew \cdot \color{blue}{\left(t \cdot t\right)}\right)\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))))))