Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.4s

Alternatives: 11

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(eh * tan(t)) / Float64(-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: 98.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   (/ (* eh (sin t)) (- (cos 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 * sin(t)) / -cos(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 * sin(t)) / Float64(-cos(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[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] / (-N[Cos[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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in eh around -inf

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

   1. lower-cos.f64 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 98.5

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

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

Alternative 3: 98.2% accurate, 1.5× speedup

Math

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in eh around -inf

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

   1. lower-cos.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 4: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in eh around -inf

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

   1. lower-cos.f64 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

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

Alternative 5: 97.7% accurate, 2.6× speedup

Math

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in eh around -inf

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

   1. lower-cos.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 98.1%

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

Alternative 6: 73.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -6.8 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.4 \cdot 10^{-21}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -6.8e+29) t_1 (if (<= eh 6.4e-21) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -6.8e+29) {
		tmp = t_1;
	} else if (eh <= 6.4e-21) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (eh <= (-6.8d+29)) then
        tmp = t_1
    else if (eh <= 6.4d-21) then
        tmp = abs((ew * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (eh <= -6.8e+29) {
		tmp = t_1;
	} else if (eh <= 6.4e-21) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -6.8e+29:
		tmp = t_1
	elif eh <= 6.4e-21:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -6.8e+29)
		tmp = t_1;
	elseif (eh <= 6.4e-21)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -6.8e+29)
		tmp = t_1;
	elseif (eh <= 6.4e-21)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -6.8e+29], t$95$1, If[LessEqual[eh, 6.4e-21], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.79999999999999963e29 or 6.4000000000000003e-21 < eh

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 52.4%

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

   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 75.8

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

   7. Applied rewrites 75.8%

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

   if -6.79999999999999963e29 < eh < 6.4000000000000003e-21

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in ew around 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 87.3

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

   7. Applied rewrites 87.3%

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

Alternative 7: 56.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -1.35 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot t, 0.5 \cdot \frac{eh \cdot eh}{ew}, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -1.35e-76)
     t_1
     (if (<= eh 3.3e-21)
       (fabs (fma (* t t) (* 0.5 (/ (* eh eh) ew)) ew))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -1.35e-76) {
		tmp = t_1;
	} else if (eh <= 3.3e-21) {
		tmp = fabs(fma((t * t), (0.5 * ((eh * eh) / ew)), ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -1.35e-76)
		tmp = t_1;
	elseif (eh <= 3.3e-21)
		tmp = abs(fma(Float64(t * t), Float64(0.5 * Float64(Float64(eh * eh) / ew)), 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[eh, -1.35e-76], t$95$1, If[LessEqual[eh, 3.3e-21], N[Abs[N[(N[(t * t), $MachinePrecision] * N[(0.5 * N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.35e-76 or 3.30000000000000009e-21 < eh

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 55.6%

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

   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 72.3

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

   7. Applied rewrites 72.3%

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

   if -1.35e-76 < eh < 3.30000000000000009e-21

   1. Initial program 99.8%

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.6

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

   7. Applied rewrites 55.6%

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

   8. Taylor expanded in ew around 0

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

   10. Applied rewrites 57.3%

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

Alternative 8: 39.0% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|t \cdot eh\right|\\ \mathbf{if}\;eh \leq -0.0042:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.4 \cdot 10^{-21}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot t, 0.5 \cdot \frac{eh \cdot eh}{ew}, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* t eh))))
   (if (<= eh -0.0042)
     t_1
     (if (<= eh 6.4e-21)
       (fabs (fma (* t t) (* 0.5 (/ (* eh eh) ew)) ew))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((t * eh));
	double tmp;
	if (eh <= -0.0042) {
		tmp = t_1;
	} else if (eh <= 6.4e-21) {
		tmp = fabs(fma((t * t), (0.5 * ((eh * eh) / ew)), ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(t * eh))
	tmp = 0.0
	if (eh <= -0.0042)
		tmp = t_1;
	elseif (eh <= 6.4e-21)
		tmp = abs(fma(Float64(t * t), Float64(0.5 * Float64(Float64(eh * eh) / ew)), ew));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -0.0042], t$95$1, If[LessEqual[eh, 6.4e-21], N[Abs[N[(N[(t * t), $MachinePrecision] * N[(0.5 * N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -0.00419999999999999974 or 6.4000000000000003e-21 < eh

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 57.8%

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

   6. Applied rewrites 26.9%

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

   7. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 75.1

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

   9. Applied rewrites 75.1%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 35.7%

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

   if -0.00419999999999999974 < eh < 6.4000000000000003e-21

   1. Initial program 99.8%

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 53.0

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

   7. Applied rewrites 53.0%

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

   8. Taylor expanded in ew around 0

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

   10. Applied rewrites 54.5%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -0.0042:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 6.4 \cdot 10^{-21}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot t, 0.5 \cdot \frac{eh \cdot eh}{ew}, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -0.0042:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -0.0042) (fabs (* t eh)) (fabs (fma (* t (* ew -0.5)) t ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -0.0042) {
		tmp = fabs((t * eh));
	} else {
		tmp = fabs(fma((t * (ew * -0.5)), t, ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -0.0042)
		tmp = abs(Float64(t * eh));
	else
		tmp = abs(fma(Float64(t * Float64(ew * -0.5)), t, ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -0.0042], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -0.00419999999999999974

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 57.9%

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 79.5

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

   9. Applied rewrites 79.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 43.0%

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

   if -0.00419999999999999974 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 39.8

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

   7. Applied rewrites 39.8%

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

   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 42.6%

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

   12. Applied rewrites 42.6%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -0.0042:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -0.0042:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-0.5, ew \cdot \left(t \cdot t\right), ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -0.0042) (fabs (* t eh)) (fabs (fma -0.5 (* ew (* t t)) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -0.0042) {
		tmp = fabs((t * eh));
	} else {
		tmp = fabs(fma(-0.5, (ew * (t * t)), ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -0.0042)
		tmp = abs(Float64(t * eh));
	else
		tmp = abs(fma(-0.5, Float64(ew * Float64(t * t)), ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -0.0042], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -0.00419999999999999974

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 57.9%

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 79.5

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

   9. Applied rewrites 79.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 43.0%

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

   if -0.00419999999999999974 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in 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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 39.8

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

   7. Applied rewrites 39.8%

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

   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 42.6%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -0.0042:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-0.5, ew \cdot \left(t \cdot t\right), ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 19.1% accurate, 107.8× speedup

Math

\[\begin{array}{l} \\ \left|t \cdot eh\right| \end{array} \]

FPCore

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

C

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

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((t * eh))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 76.5%

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

6. Applied rewrites 37.3%

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

7. Taylor expanded in ew around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 46.6

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

9. Applied rewrites 46.6%

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

10. Taylor expanded in t around 0

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

12. Applied rewrites 22.3%

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

13. Final simplification 22.3%

    \[\leadsto \left|t \cdot eh\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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)))))))