Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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]]

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. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eh around 0

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

   1. lower-cos.f64 97.6

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

7. Simplified 97.6%

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

8. Final simplification 97.6%

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

Alternative 4: 75.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(eh \cdot 0.5\right) \cdot \left(\sin t \cdot \frac{\tan t}{ew}\right)\\ \mathbf{if}\;ew \leq -1.6 \cdot 10^{-89}:\\ \;\;\;\;\left|\mathsf{fma}\left(t\_1, eh, ew \cdot \cos t\right)\right|\\ \mathbf{elif}\;ew \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, eh \cdot t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (* eh 0.5) (* (sin t) (/ (tan t) ew)))))
   (if (<= ew -1.6e-89)
     (fabs (fma t_1 eh (* ew (cos t))))
     (if (<= ew 1.2e-36)
       (fabs (* eh (sin t)))
       (fabs (fma (cos t) ew (* eh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * 0.5) * (sin(t) * (tan(t) / ew));
	double tmp;
	if (ew <= -1.6e-89) {
		tmp = fabs(fma(t_1, eh, (ew * cos(t))));
	} else if (ew <= 1.2e-36) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = fabs(fma(cos(t), ew, (eh * t_1)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * 0.5) * Float64(sin(t) * Float64(tan(t) / ew)))
	tmp = 0.0
	if (ew <= -1.6e-89)
		tmp = abs(fma(t_1, eh, Float64(ew * cos(t))));
	elseif (ew <= 1.2e-36)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = abs(fma(cos(t), ew, Float64(eh * t_1)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * 0.5), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.6e-89], N[Abs[N[(t$95$1 * eh + N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.2e-36], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew + N[(eh * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.59999999999999999e-89

   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 89.6%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 71.0

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

   7. Simplified 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied egg-rr 80.8%

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

   if -1.59999999999999999e-89 < ew < 1.2e-36

   1. Initial program 99.7%

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

   4. Applied egg-rr 62.4%

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

   5. Taylor expanded in ew around 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 72.2

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

   7. Simplified 72.2%

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

   if 1.2e-36 < 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 91.4%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 74.7

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

   7. Simplified 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

   9. Applied egg-rr 86.5%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos t, ew, eh \cdot \left(\left(eh \cdot 0.5\right) \cdot \left(\sin t \cdot \frac{\tan t}{ew}\right)\right)\right)}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.6 \cdot 10^{-89}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(eh \cdot 0.5\right) \cdot \left(\sin t \cdot \frac{\tan t}{ew}\right), eh, ew \cdot \cos t\right)\right|\\ \mathbf{elif}\;ew \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, eh \cdot \left(\left(eh \cdot 0.5\right) \cdot \left(\sin t \cdot \frac{\tan t}{ew}\right)\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\mathsf{fma}\left(\cos t, ew, eh \cdot \left(\left(eh \cdot 0.5\right) \cdot \left(\sin t \cdot \frac{\tan t}{ew}\right)\right)\right)\right|\\ \mathbf{if}\;ew \leq -1.6 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (fma (cos t) ew (* eh (* (* eh 0.5) (* (sin t) (/ (tan t) ew))))))))
   (if (<= ew -1.6e-89) t_1 (if (<= ew 1.2e-36) (fabs (* eh (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(fma(cos(t), ew, (eh * ((eh * 0.5) * (sin(t) * (tan(t) / ew))))));
	double tmp;
	if (ew <= -1.6e-89) {
		tmp = t_1;
	} else if (ew <= 1.2e-36) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(fma(cos(t), ew, Float64(eh * Float64(Float64(eh * 0.5) * Float64(sin(t) * Float64(tan(t) / ew))))))
	tmp = 0.0
	if (ew <= -1.6e-89)
		tmp = t_1;
	elseif (ew <= 1.2e-36)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[t], $MachinePrecision] * ew + N[(eh * N[(N[(eh * 0.5), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.6e-89], t$95$1, If[LessEqual[ew, 1.2e-36], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.59999999999999999e-89 or 1.2e-36 < 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 90.5%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 72.9

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

   7. Simplified 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

   9. Applied egg-rr 83.7%

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

   if -1.59999999999999999e-89 < ew < 1.2e-36

   1. Initial program 99.7%

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

   4. Applied egg-rr 62.4%

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

   5. Taylor expanded in ew around 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 72.2

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

   7. Simplified 72.2%

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

Alternative 6: 75.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -1.6 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.2 \cdot 10^{-36}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -1.6e-89) t_1 (if (<= ew 4.2e-36) (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 <= -1.6e-89) {
		tmp = t_1;
	} else if (ew <= 4.2e-36) {
		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 <= (-1.6d-89)) then
        tmp = t_1
    else if (ew <= 4.2d-36) 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 <= -1.6e-89) {
		tmp = t_1;
	} else if (ew <= 4.2e-36) {
		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 <= -1.6e-89:
		tmp = t_1
	elif ew <= 4.2e-36:
		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 <= -1.6e-89)
		tmp = t_1;
	elseif (ew <= 4.2e-36)
		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 <= -1.6e-89)
		tmp = t_1;
	elseif (ew <= 4.2e-36)
		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, -1.6e-89], t$95$1, If[LessEqual[ew, 4.2e-36], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.59999999999999999e-89 or 4.19999999999999982e-36 < 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 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 83.2

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

   7. Simplified 83.2%

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

   if -1.59999999999999999e-89 < ew < 4.19999999999999982e-36

   1. Initial program 99.7%

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

   4. Applied egg-rr 62.4%

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

   5. Taylor expanded in ew around 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 72.2

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

   7. Simplified 72.2%

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

Alternative 7: 61.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -5e-6)
     t_1
     (if (<= t 1.5e-43) (fabs (fma (* t (* ew -0.5)) t ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -5e-6) {
		tmp = t_1;
	} else if (t <= 1.5e-43) {
		tmp = fabs(fma((t * (ew * -0.5)), 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 <= -5e-6)
		tmp = t_1;
	elseif (t <= 1.5e-43)
		tmp = abs(fma(Float64(t * Float64(ew * -0.5)), 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, -5e-6], t$95$1, If[LessEqual[t, 1.5e-43], N[Abs[N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.00000000000000041e-6 or 1.50000000000000002e-43 < t

   1. Initial program 99.6%

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

   4. Applied egg-rr 69.7%

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

   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 51.8

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

   7. Simplified 51.8%

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

   if -5.00000000000000041e-6 < t < 1.50000000000000002e-43

   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 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 t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 66.0

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

   7. Simplified 66.0%

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

   8. Taylor expanded in ew around inf

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

      1. lower-*.f64 76.5

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

   10. Simplified 76.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 76.5

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 76.5

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

   12. Applied egg-rr 76.5%

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

Alternative 8: 38.1% accurate, 45.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied egg-rr 79.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 33.0

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

7. Simplified 33.0%

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

8. Taylor expanded in ew around inf

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

   1. lower-*.f64 39.1

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

10. Simplified 39.1%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-*.f64 39.3

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 39.3

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

12. Applied egg-rr 39.3%

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

Alternative 9: 38.1% accurate, 45.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied egg-rr 79.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 33.0

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

7. Simplified 33.0%

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

8. Taylor expanded in ew around inf

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

   1. lower-*.f64 39.1

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

10. Simplified 39.1%

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

11. Final simplification 39.1%

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

Alternative 10: 4.8% accurate, 47.9× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew * (-0.5 * (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((ew * ((-0.5d0) * (t * t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied egg-rr 79.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 33.0

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

7. Simplified 33.0%

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

8. Taylor expanded in ew around inf

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

   1. lower-*.f64 39.1

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

10. Simplified 39.1%

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

11. Taylor expanded in t around inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 4.9

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

13. Simplified 4.9%

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

Reproduce

herbie shell --seed 2024208 
(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)))))))