Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.4s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 81.5%

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

5. Taylor expanded in eh around -inf

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

   1. cos-lowering-cos.f64 98.4

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

7. Simplified 98.4%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

9. Applied egg-rr 98.4%

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

10. Taylor expanded in t around inf

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

    1. mul-1-neg N/A

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

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

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

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

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

    4. sin-lowering-sin.f64 98.5

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 3: 98.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 81.5%

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

5. Taylor expanded in eh around -inf

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

   1. cos-lowering-cos.f64 98.4

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

7. Simplified 98.4%

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

8. Taylor expanded in eh around 0

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

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

Alternative 4: 86.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\mathsf{fma}\left(\tan t \cdot \left(-eh\right), \cos t, ew\right)\right|\\ \mathbf{if}\;eh \leq -9500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.3 \cdot 10^{-85}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (fma (* (tan t) (- eh)) (cos t) ew))))
   (if (<= eh -9500000.0) t_1 (if (<= eh 3.3e-85) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(fma((tan(t) * -eh), cos(t), ew));
	double tmp;
	if (eh <= -9500000.0) {
		tmp = t_1;
	} else if (eh <= 3.3e-85) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(fma(Float64(tan(t) * Float64(-eh)), cos(t), ew))
	tmp = 0.0
	if (eh <= -9500000.0)
		tmp = t_1;
	elseif (eh <= 3.3e-85)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Tan[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Cos[t], $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -9500000.0], t$95$1, If[LessEqual[eh, 3.3e-85], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -9.5e6 or 3.29999999999999973e-85 < 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. sub-neg N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 67.5%

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

   5. Taylor expanded in eh around -inf

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

      1. cos-lowering-cos.f64 97.8

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

   7. Simplified 97.8%

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

   8. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. neg-lowering-neg.f64 92.0

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

   10. Simplified 92.0%

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

   11. Taylor expanded in t around 0

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

   13. Simplified 87.3%

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

   if -9.5e6 < eh < 3.29999999999999973e-85

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in ew around inf

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

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

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

      2. cos-lowering-cos.f64 91.8

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

   7. Simplified 91.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9500000:\\ \;\;\;\;\left|\mathsf{fma}\left(\tan t \cdot \left(-eh\right), \cos t, ew\right)\right|\\ \mathbf{elif}\;eh \leq 3.3 \cdot 10^{-85}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\tan t \cdot \left(-eh\right), \cos t, ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{+118}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;t \leq -0.00017:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.0062:\\ \;\;\;\;\left|eh \cdot t - ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= t -7.4e+118)
     (fabs (* eh (sin t)))
     (if (<= t -0.00017) t_1 (if (<= t 0.0062) (fabs (- (* eh t) ew)) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (t <= -7.4e+118) {
		tmp = fabs((eh * sin(t)));
	} else if (t <= -0.00017) {
		tmp = t_1;
	} else if (t <= 0.0062) {
		tmp = fabs(((eh * t) - ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (t <= (-7.4d+118)) then
        tmp = abs((eh * sin(t)))
    else if (t <= (-0.00017d0)) then
        tmp = t_1
    else if (t <= 0.0062d0) then
        tmp = abs(((eh * t) - ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (t <= -7.4e+118) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else if (t <= -0.00017) {
		tmp = t_1;
	} else if (t <= 0.0062) {
		tmp = Math.abs(((eh * t) - ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if t <= -7.4e+118:
		tmp = math.fabs((eh * math.sin(t)))
	elif t <= -0.00017:
		tmp = t_1
	elif t <= 0.0062:
		tmp = math.fabs(((eh * t) - ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (t <= -7.4e+118)
		tmp = abs(Float64(eh * sin(t)));
	elseif (t <= -0.00017)
		tmp = t_1;
	elseif (t <= 0.0062)
		tmp = abs(Float64(Float64(eh * t) - ew));
	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 (t <= -7.4e+118)
		tmp = abs((eh * sin(t)));
	elseif (t <= -0.00017)
		tmp = t_1;
	elseif (t <= 0.0062)
		tmp = abs(((eh * t) - ew));
	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[t, -7.4e+118], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, -0.00017], t$95$1, If[LessEqual[t, 0.0062], N[Abs[N[(N[(eh * t), $MachinePrecision] - ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.39999999999999973e118

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 55.5%

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

   5. Taylor expanded in eh around inf

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

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

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

      2. sin-lowering-sin.f64 65.8

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

   7. Simplified 65.8%

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

   if -7.39999999999999973e118 < t < -1.7e-4 or 0.00619999999999999978 < t

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in ew around inf

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

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

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

      2. cos-lowering-cos.f64 64.0

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

   7. Simplified 64.0%

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

   if -1.7e-4 < t < 0.00619999999999999978

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 90.0%

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

   5. Taylor expanded in eh around -inf

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

      1. cos-lowering-cos.f64 98.9

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

   7. Simplified 98.9%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left|ew + \color{blue}{\left(\mathsf{neg}\left(eh \cdot t\right)\right)}\right| \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 98.3

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

   10. Simplified 98.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{+118}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;t \leq -0.00017:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.0062:\\ \;\;\;\;\left|eh \cdot t - ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -48000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-18}:\\ \;\;\;\;\left|eh \cdot t - ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -48000000.0)
     t_1
     (if (<= t 1.75e-18) (fabs (- (* eh t) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -48000000.0) {
		tmp = t_1;
	} else if (t <= 1.75e-18) {
		tmp = fabs(((eh * t) - ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (t <= (-48000000.0d0)) then
        tmp = t_1
    else if (t <= 1.75d-18) then
        tmp = abs(((eh * t) - ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (t <= -48000000.0) {
		tmp = t_1;
	} else if (t <= 1.75e-18) {
		tmp = Math.abs(((eh * t) - ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if t <= -48000000.0:
		tmp = t_1
	elif t <= 1.75e-18:
		tmp = math.fabs(((eh * 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 <= -48000000.0)
		tmp = t_1;
	elseif (t <= 1.75e-18)
		tmp = abs(Float64(Float64(eh * t) - ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (t <= -48000000.0)
		tmp = t_1;
	elseif (t <= 1.75e-18)
		tmp = abs(((eh * t) - ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -48000000.0], t$95$1, If[LessEqual[t, 1.75e-18], N[Abs[N[(N[(eh * t), $MachinePrecision] - ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8e7 or 1.7499999999999999e-18 < t

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 72.7%

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

   5. Taylor expanded in eh around inf

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

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

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

      2. sin-lowering-sin.f64 47.6

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

   7. Simplified 47.6%

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

   if -4.8e7 < t < 1.7499999999999999e-18

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in eh around -inf

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

      1. cos-lowering-cos.f64 99.6

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

   7. Simplified 99.6%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left|ew + \color{blue}{\left(\mathsf{neg}\left(eh \cdot t\right)\right)}\right| \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 98.3

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

   10. Simplified 98.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -48000000:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-18}:\\ \;\;\;\;\left|eh \cdot t - ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6600000000:\\ \;\;\;\;\left|eh \cdot t - ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))))
   (if (<= t -3.1e+25)
     t_1
     (if (<= t 6600000000.0) (fabs (- (* eh t) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double tmp;
	if (t <= -3.1e+25) {
		tmp = t_1;
	} else if (t <= 6600000000.0) {
		tmp = fabs(((eh * t) - ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ew * cos(t)
    if (t <= (-3.1d+25)) then
        tmp = t_1
    else if (t <= 6600000000.0d0) then
        tmp = abs(((eh * t) - ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double tmp;
	if (t <= -3.1e+25) {
		tmp = t_1;
	} else if (t <= 6600000000.0) {
		tmp = Math.abs(((eh * t) - ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	tmp = 0
	if t <= -3.1e+25:
		tmp = t_1
	elif t <= 6600000000.0:
		tmp = math.fabs(((eh * t) - ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	tmp = 0.0
	if (t <= -3.1e+25)
		tmp = t_1;
	elseif (t <= 6600000000.0)
		tmp = abs(Float64(Float64(eh * t) - ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	tmp = 0.0;
	if (t <= -3.1e+25)
		tmp = t_1;
	elseif (t <= 6600000000.0)
		tmp = abs(((eh * t) - ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.1e+25], t$95$1, If[LessEqual[t, 6600000000.0], N[Abs[N[(N[(eh * t), $MachinePrecision] - ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.0999999999999998e25 or 6.6e9 < t

   1. Initial program 99.8%

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

   4. Applied egg-rr 67.6%

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

   5. Taylor expanded in ew around inf

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

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

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

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

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

      3. cos-lowering-cos.f64 57.0

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

   7. Simplified 57.0%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. fabs-sqr N/A

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

      4. sqr-pow N/A

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

      5. inv-pow N/A

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

      6. clear-num N/A

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

      7. /-rgt-identity N/A

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

      8. *-commutative N/A

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

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

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

      10. cos-lowering-cos.f64 26.6

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

   9. Applied egg-rr 26.6%

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

   if -3.0999999999999998e25 < t < 6.6e9

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in eh around -inf

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

      1. cos-lowering-cos.f64 99.0

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

   7. Simplified 99.0%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left|ew + \color{blue}{\left(\mathsf{neg}\left(eh \cdot t\right)\right)}\right| \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 91.7

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

   10. Simplified 91.7%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;ew \cdot \cos t\\ \mathbf{elif}\;t \leq 6600000000:\\ \;\;\;\;\left|eh \cdot t - ew\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.4% accurate, 61.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 4.2 \cdot 10^{+177}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 4.2e+177) (fabs ew) (fabs (* eh t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 4.2e+177) {
		tmp = fabs(ew);
	} else {
		tmp = fabs((eh * t));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= 4.2d+177) then
        tmp = abs(ew)
    else
        tmp = abs((eh * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 4.2e+177) {
		tmp = Math.abs(ew);
	} else {
		tmp = Math.abs((eh * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 4.2e+177:
		tmp = math.fabs(ew)
	else:
		tmp = math.fabs((eh * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 4.2e+177)
		tmp = abs(ew);
	else
		tmp = abs(Float64(eh * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 4.2e+177)
		tmp = abs(ew);
	else
		tmp = abs((eh * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 4.2e+177], N[Abs[ew], $MachinePrecision], N[Abs[N[(eh * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 4.20000000000000026e177

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0

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

   7. Simplified 44.4%

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

   if 4.20000000000000026e177 < 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. sub-neg N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 56.2%

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

   5. Taylor expanded in eh around -inf

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

      1. cos-lowering-cos.f64 96.9

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

   7. Simplified 96.9%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left|ew + \color{blue}{\left(\mathsf{neg}\left(eh \cdot t\right)\right)}\right| \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 65.1

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

   10. Simplified 65.1%

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

   11. Taylor expanded in ew around 0

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

       1. mul-1-neg N/A

          \[\leadsto \left|\color{blue}{\mathsf{neg}\left(eh \cdot t\right)}\right| \]

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

          \[\leadsto \left|\color{blue}{\mathsf{neg}\left(eh \cdot t\right)}\right| \]

       3. *-lowering-*.f64 48.2

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

   13. Simplified 48.2%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 4.2 \cdot 10^{+177}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.9% accurate, 78.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 81.5%

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

5. Taylor expanded in eh around -inf

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

   1. cos-lowering-cos.f64 98.4

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

7. Simplified 98.4%

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

8. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

      \[\leadsto \left|ew + \color{blue}{\left(\mathsf{neg}\left(eh \cdot t\right)\right)}\right| \]

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 55.3

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

10. Simplified 55.3%

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

11. Final simplification 55.3%

    \[\leadsto \left|eh \cdot t - ew\right| \]
12. Add Preprocessing

Alternative 10: 42.2% accurate, 287.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in t around 0

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

7. Simplified 42.2%

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

Reproduce

herbie shell --seed 2024204 
(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)))))))