Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.5s

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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh \cdot \tan t}{0 - ew}\\ \left|\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {t\_1}^{2}}}, ew, eh \cdot \left(\sin \tan^{-1} t\_1 \cdot \left(0 - \sin t\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* eh (tan t)) (- 0.0 ew))))
   (fabs
    (fma
     (/ (cos t) (sqrt (+ 1.0 (pow t_1 2.0))))
     ew
     (* eh (* (sin (atan t_1)) (- 0.0 (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * tan(t)) / (0.0 - ew);
	return fabs(fma((cos(t) / sqrt((1.0 + pow(t_1, 2.0)))), ew, (eh * (sin(atan(t_1)) * (0.0 - sin(t))))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * tan(t)) / Float64(0.0 - ew))
	return abs(fma(Float64(cos(t) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))), ew, Float64(eh * Float64(sin(atan(t_1)) * Float64(0.0 - sin(t))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Cos[t], $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(eh * N[(N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sin[t], $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. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}, ew, \left(0 - eh\right) \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right)}\right| \]

5. Final simplification 99.8%

   \[\leadsto \left|\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}, ew, eh \cdot \left(\sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right) \cdot \left(0 - \sin t\right)\right)\right)\right| \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh \cdot \tan t}{0 - ew}\\ \left|\mathsf{fma}\left(\sin t, \left(0 - eh\right) \cdot \sin \tan^{-1} t\_1, \frac{\cos t \cdot ew}{\sqrt{1 + {t\_1}^{2}}}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* eh (tan t)) (- 0.0 ew))))
   (fabs
    (fma
     (sin t)
     (* (- 0.0 eh) (sin (atan t_1)))
     (/ (* (cos t) ew) (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * tan(t)) / (0.0 - ew);
	return fabs(fma(sin(t), ((0.0 - eh) * sin(atan(t_1))), ((cos(t) * ew) / sqrt((1.0 + pow(t_1, 2.0))))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * tan(t)) / Float64(0.0 - ew))
	return abs(fma(sin(t), Float64(Float64(0.0 - eh) * sin(atan(t_1))), Float64(Float64(cos(t) * ew) / sqrt(Float64(1.0 + (t_1 ^ 2.0))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(N[(0.0 - eh), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

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

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

   6. *-commutative N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\sin t, -eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)}\right| \]

5. Final simplification 99.8%

   \[\leadsto \left|\mathsf{fma}\left(\sin t, \left(0 - eh\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right), \frac{\cos t \cdot ew}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]
6. Add Preprocessing

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(\sin t, eh \cdot \left(eh \cdot \frac{-1}{eh}\right), \frac{\cos t \cdot ew}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   (sin t)
   (* eh (* eh (/ -1.0 eh)))
   (/ (* (cos t) ew) (sqrt (+ 1.0 (pow (/ (* eh (tan t)) (- 0.0 ew)) 2.0)))))))

C

double code(double eh, double ew, double t) {
	return fabs(fma(sin(t), (eh * (eh * (-1.0 / eh))), ((cos(t) * ew) / sqrt((1.0 + pow(((eh * tan(t)) / (0.0 - ew)), 2.0))))));
}

Julia

function code(eh, ew, t)
	return abs(fma(sin(t), Float64(eh * Float64(eh * Float64(-1.0 / eh))), Float64(Float64(cos(t) * ew) / sqrt(Float64(1.0 + (Float64(Float64(eh * tan(t)) / Float64(0.0 - ew)) ^ 2.0))))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(eh * N[(eh * N[(-1.0 / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - 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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

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

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

   6. *-commutative N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\sin t, -eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)}\right| \]
5. Step-by-step derivation

   1. sin-atan N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \color{blue}{\frac{\frac{eh \cdot \tan t}{0 - ew}}{\sqrt{1 + \frac{eh \cdot \tan t}{0 - ew} \cdot \frac{eh \cdot \tan t}{0 - ew}}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   2. associate-/l* N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{\color{blue}{eh \cdot \frac{\tan t}{0 - ew}}}{\sqrt{1 + \frac{eh \cdot \tan t}{0 - ew} \cdot \frac{eh \cdot \tan t}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   3. *-commutative N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \frac{\color{blue}{\tan t \cdot eh}}{0 - ew} \cdot \frac{eh \cdot \tan t}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   4. associate-*r/ N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew}\right)} \cdot \frac{eh \cdot \tan t}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   5. +-rgt-identity N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew} + 0\right)} \cdot \frac{eh \cdot \tan t}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   6. *-commutative N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{0 - ew} + 0\right) \cdot \frac{\color{blue}{\tan t \cdot eh}}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   7. associate-*r/ N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{0 - ew} + 0\right) \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew}\right)}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   8. +-rgt-identity N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{0 - ew} + 0\right) \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew} + 0\right)}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   9. +-rgt-identity N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew}\right)} \cdot \left(\tan t \cdot \frac{eh}{0 - ew} + 0\right)}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   10. +-rgt-identity N/A

       \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{0 - ew}\right) \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew}\right)}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

6. Applied egg-rr 84.1%

   \[\leadsto \left|\mathsf{fma}\left(\sin t, -eh \cdot \color{blue}{\left(eh \cdot \frac{\frac{0 - \tan t}{ew}}{\sqrt{1 + {\left(\mathsf{fma}\left(eh, \frac{0 - \tan t}{ew}, 0\right)\right)}^{2}}}\right)}, \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

7. Taylor expanded in eh around -inf

   \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \left(eh \cdot \color{blue}{\frac{1}{eh}}\right)\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]
8. Step-by-step derivation

   1. /-lowering-/.f64 97.8

      \[\leadsto \left|\mathsf{fma}\left(\sin t, -eh \cdot \left(eh \cdot \color{blue}{\frac{1}{eh}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

9. Simplified 97.8%

   \[\leadsto \left|\mathsf{fma}\left(\sin t, -eh \cdot \left(eh \cdot \color{blue}{\frac{1}{eh}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

10. Final simplification 97.8%

    \[\leadsto \left|\mathsf{fma}\left(\sin t, eh \cdot \left(eh \cdot \frac{-1}{eh}\right), \frac{\cos t \cdot ew}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]
11. Add Preprocessing

Alternative 4: 89.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \left(\cos t - eh \cdot \frac{\sin t}{ew}\right)\right| \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew * (cos(t) - (eh * (sin(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((ew * (cos(t) - (eh * (sin(t) / ew)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

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

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

   6. *-commutative N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\sin t, -eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)}\right| \]
5. Step-by-step derivation

   1. sin-atan N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \color{blue}{\frac{\frac{eh \cdot \tan t}{0 - ew}}{\sqrt{1 + \frac{eh \cdot \tan t}{0 - ew} \cdot \frac{eh \cdot \tan t}{0 - ew}}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   2. associate-/l* N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{\color{blue}{eh \cdot \frac{\tan t}{0 - ew}}}{\sqrt{1 + \frac{eh \cdot \tan t}{0 - ew} \cdot \frac{eh \cdot \tan t}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   3. *-commutative N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \frac{\color{blue}{\tan t \cdot eh}}{0 - ew} \cdot \frac{eh \cdot \tan t}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   4. associate-*r/ N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew}\right)} \cdot \frac{eh \cdot \tan t}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   5. +-rgt-identity N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew} + 0\right)} \cdot \frac{eh \cdot \tan t}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   6. *-commutative N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{0 - ew} + 0\right) \cdot \frac{\color{blue}{\tan t \cdot eh}}{0 - ew}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   7. associate-*r/ N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{0 - ew} + 0\right) \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew}\right)}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   8. +-rgt-identity N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{0 - ew} + 0\right) \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew} + 0\right)}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   9. +-rgt-identity N/A

      \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew}\right)} \cdot \left(\tan t \cdot \frac{eh}{0 - ew} + 0\right)}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

   10. +-rgt-identity N/A

       \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \frac{eh \cdot \frac{\tan t}{0 - ew}}{\sqrt{1 + \left(\tan t \cdot \frac{eh}{0 - ew}\right) \cdot \color{blue}{\left(\tan t \cdot \frac{eh}{0 - ew}\right)}}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

6. Applied egg-rr 84.1%

   \[\leadsto \left|\mathsf{fma}\left(\sin t, -eh \cdot \color{blue}{\left(eh \cdot \frac{\frac{0 - \tan t}{ew}}{\sqrt{1 + {\left(\mathsf{fma}\left(eh, \frac{0 - \tan t}{ew}, 0\right)\right)}^{2}}}\right)}, \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

7. Taylor expanded in eh around -inf

   \[\leadsto \left|\mathsf{fma}\left(\sin t, \mathsf{neg}\left(eh \cdot \left(eh \cdot \color{blue}{\frac{1}{eh}}\right)\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]
8. Step-by-step derivation

   1. /-lowering-/.f64 97.8

      \[\leadsto \left|\mathsf{fma}\left(\sin t, -eh \cdot \left(eh \cdot \color{blue}{\frac{1}{eh}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

9. Simplified 97.8%

   \[\leadsto \left|\mathsf{fma}\left(\sin t, -eh \cdot \left(eh \cdot \color{blue}{\frac{1}{eh}}\right), \frac{ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right)\right| \]

10. Taylor expanded in ew around inf

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

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

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

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

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

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

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

    6. associate-/l* N/A

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

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

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

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

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

    9. sin-lowering-sin.f64 90.0

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

12. Simplified 90.0%

    \[\leadsto \left|\color{blue}{ew \cdot \left(\cos t - eh \cdot \frac{\sin t}{ew}\right)}\right| \]
13. Add Preprocessing

Alternative 5: 74.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos t \cdot ew\right|\\ \mathbf{if}\;ew \leq -1.62 \cdot 10^{-93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.25 \cdot 10^{-101}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (cos t) ew))))
   (if (<= ew -1.62e-93)
     t_1
     (if (<= ew 1.25e-101) (fabs (* eh (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(t) * ew));
	double tmp;
	if (ew <= -1.62e-93) {
		tmp = t_1;
	} else if (ew <= 1.25e-101) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((cos(t) * ew))
    if (ew <= (-1.62d-93)) then
        tmp = t_1
    else if (ew <= 1.25d-101) then
        tmp = abs((eh * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.cos(t) * ew));
	double tmp;
	if (ew <= -1.62e-93) {
		tmp = t_1;
	} else if (ew <= 1.25e-101) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(t) * ew))
	tmp = 0
	if ew <= -1.62e-93:
		tmp = t_1
	elif ew <= 1.25e-101:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(t) * ew))
	tmp = 0.0
	if (ew <= -1.62e-93)
		tmp = t_1;
	elseif (ew <= 1.25e-101)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(t) * ew));
	tmp = 0.0;
	if (ew <= -1.62e-93)
		tmp = t_1;
	elseif (ew <= 1.25e-101)
		tmp = abs((eh * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.62e-93], t$95$1, If[LessEqual[ew, 1.25e-101], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.6200000000000001e-93 or 1.25e-101 < ew

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}, ew, \left(0 - eh\right) \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right)}\right| \]

   5. Taylor expanded in eh around 0

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

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

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

      2. cos-lowering-cos.f64 81.6

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

   7. Simplified 81.6%

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

   if -1.6200000000000001e-93 < ew < 1.25e-101

   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. neg-fabs N/A

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{\left|\frac{\left(0 - eh\right) \cdot \left(\frac{\tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right) - ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right|} \]

   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 71.1

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

   7. Simplified 71.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.62 \cdot 10^{-93}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;ew \leq 1.25 \cdot 10^{-101}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -0.00016:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-97}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -0.00016) t_1 (if (<= t 2.4e-97) (fabs ew) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -0.00016) {
		tmp = t_1;
	} else if (t <= 2.4e-97) {
		tmp = fabs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (t <= (-0.00016d0)) then
        tmp = t_1
    else if (t <= 2.4d-97) then
        tmp = abs(ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (t <= -0.00016) {
		tmp = t_1;
	} else if (t <= 2.4e-97) {
		tmp = Math.abs(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 <= -0.00016:
		tmp = t_1
	elif t <= 2.4e-97:
		tmp = math.fabs(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 <= -0.00016)
		tmp = t_1;
	elseif (t <= 2.4e-97)
		tmp = abs(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 <= -0.00016)
		tmp = t_1;
	elseif (t <= 2.4e-97)
		tmp = abs(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, -0.00016], t$95$1, If[LessEqual[t, 2.4e-97], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.60000000000000013e-4 or 2.4e-97 < t

   1. Initial program 99.6%

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

      1. neg-fabs N/A

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\left|\frac{\left(0 - eh\right) \cdot \left(\frac{\tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right) - ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right|} \]

   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 45.6

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

   7. Simplified 45.6%

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

   if -1.60000000000000013e-4 < t < 2.4e-97

   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. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}, ew, \left(0 - eh\right) \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right)}\right| \]

   5. Taylor expanded in t around 0

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

   7. Simplified 79.4%

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

Alternative 7: 44.5% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.08 \cdot 10^{-148}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 3.4 \cdot 10^{-221}:\\ \;\;\;\;\left|eh \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.08e-148)
   (fabs ew)
   (if (<= ew 3.4e-221)
     (fabs
      (*
       eh
       (*
        t
        (fma
         (* t t)
         (fma (* t t) 0.008333333333333333 -0.16666666666666666)
         1.0))))
     (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.08e-148) {
		tmp = fabs(ew);
	} else if (ew <= 3.4e-221) {
		tmp = fabs((eh * (t * fma((t * t), fma((t * t), 0.008333333333333333, -0.16666666666666666), 1.0))));
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.08e-148)
		tmp = abs(ew);
	elseif (ew <= 3.4e-221)
		tmp = abs(Float64(eh * Float64(t * fma(Float64(t * t), fma(Float64(t * t), 0.008333333333333333, -0.16666666666666666), 1.0))));
	else
		tmp = abs(ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.08e-148], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 3.4e-221], N[Abs[N[(eh * N[(t * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.08000000000000006e-148 or 3.4000000000000001e-221 < ew

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}, ew, \left(0 - eh\right) \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right)}\right| \]

   5. Taylor expanded in t around 0

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

   7. Simplified 50.4%

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

   if -1.08000000000000006e-148 < ew < 3.4000000000000001e-221

   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. neg-fabs N/A

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\left|\frac{\left(0 - eh\right) \cdot \left(\frac{\tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right) - ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right|} \]

   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 81.1

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

   7. Simplified 81.1%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|eh \cdot \color{blue}{\left(t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)}\right| \]
   9. Step-by-step derivation

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

         \[\leadsto \left|eh \cdot \color{blue}{\left(t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)\right)}\right| \]

      2. +-commutative N/A

         \[\leadsto \left|eh \cdot \left(t \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right) + 1\right)}\right)\right| \]

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

         \[\leadsto \left|eh \cdot \left(t \cdot \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{120} \cdot {t}^{2} - \frac{1}{6}, 1\right)}\right)\right| \]

      4. unpow2 N/A

         \[\leadsto \left|eh \cdot \left(t \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{120} \cdot {t}^{2} - \frac{1}{6}, 1\right)\right)\right| \]

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

         \[\leadsto \left|eh \cdot \left(t \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{120} \cdot {t}^{2} - \frac{1}{6}, 1\right)\right)\right| \]

      6. sub-neg N/A

         \[\leadsto \left|eh \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{\frac{1}{120} \cdot {t}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right)\right| \]

      7. *-commutative N/A

         \[\leadsto \left|eh \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right)\right| \]

      8. metadata-eval N/A

         \[\leadsto \left|eh \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, {t}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, 1\right)\right)\right| \]

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

         \[\leadsto \left|eh \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)\right)\right| \]

      10. unpow2 N/A

          \[\leadsto \left|eh \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right| \]

      11. *-lowering-*.f64 43.8

          \[\leadsto \left|eh \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right| \]

   10. Simplified 43.8%

       \[\leadsto \left|eh \cdot \color{blue}{\left(t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 44.5% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -4.4 \cdot 10^{-148}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 7.8 \cdot 10^{-224}:\\ \;\;\;\;\left|t \cdot \mathsf{fma}\left(-0.16666666666666666, eh \cdot \left(t \cdot t\right), eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -4.4e-148)
   (fabs ew)
   (if (<= ew 7.8e-224)
     (fabs (* t (fma -0.16666666666666666 (* eh (* t t)) eh)))
     (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.4e-148) {
		tmp = fabs(ew);
	} else if (ew <= 7.8e-224) {
		tmp = fabs((t * fma(-0.16666666666666666, (eh * (t * t)), eh)));
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -4.4e-148)
		tmp = abs(ew);
	elseif (ew <= 7.8e-224)
		tmp = abs(Float64(t * fma(-0.16666666666666666, Float64(eh * Float64(t * t)), eh)));
	else
		tmp = abs(ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -4.4e-148], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 7.8e-224], N[Abs[N[(t * N[(-0.16666666666666666 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] + eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.40000000000000034e-148 or 7.7999999999999996e-224 < ew

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}, ew, \left(0 - eh\right) \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right)}\right| \]

   5. Taylor expanded in t around 0

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

   7. Simplified 50.4%

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

   if -4.40000000000000034e-148 < ew < 7.7999999999999996e-224

   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. neg-fabs N/A

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\left|\frac{\left(0 - eh\right) \cdot \left(\frac{\tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right) - ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right|} \]

   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 81.1

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

   7. Simplified 81.1%

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

   8. Taylor expanded in t around 0

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

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

         \[\leadsto \left|\color{blue}{t \cdot \left(eh + \frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)\right)}\right| \]

      2. +-commutative N/A

         \[\leadsto \left|t \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right) + eh\right)}\right| \]

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

         \[\leadsto \left|t \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, eh \cdot {t}^{2}, eh\right)}\right| \]

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

         \[\leadsto \left|t \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{eh \cdot {t}^{2}}, eh\right)\right| \]

      5. unpow2 N/A

         \[\leadsto \left|t \cdot \mathsf{fma}\left(\frac{-1}{6}, eh \cdot \color{blue}{\left(t \cdot t\right)}, eh\right)\right| \]

      6. *-lowering-*.f64 43.6

         \[\leadsto \left|t \cdot \mathsf{fma}\left(-0.16666666666666666, eh \cdot \color{blue}{\left(t \cdot t\right)}, eh\right)\right| \]

   10. Simplified 43.6%

       \[\leadsto \left|\color{blue}{t \cdot \mathsf{fma}\left(-0.16666666666666666, eh \cdot \left(t \cdot t\right), eh\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 44.6% accurate, 43.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -4.8 \cdot 10^{-147}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 1.15 \cdot 10^{-221}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -4.8e-147)
   (fabs ew)
   (if (<= ew 1.15e-221) (fabs (* t eh)) (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.8e-147) {
		tmp = fabs(ew);
	} else if (ew <= 1.15e-221) {
		tmp = fabs((t * eh));
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-4.8d-147)) then
        tmp = abs(ew)
    else if (ew <= 1.15d-221) then
        tmp = abs((t * eh))
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.8e-147) {
		tmp = Math.abs(ew);
	} else if (ew <= 1.15e-221) {
		tmp = Math.abs((t * eh));
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -4.8e-147:
		tmp = math.fabs(ew)
	elif ew <= 1.15e-221:
		tmp = math.fabs((t * eh))
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -4.8e-147)
		tmp = abs(ew);
	elseif (ew <= 1.15e-221)
		tmp = abs(Float64(t * eh));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -4.8e-147)
		tmp = abs(ew);
	elseif (ew <= 1.15e-221)
		tmp = abs((t * eh));
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -4.8e-147], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 1.15e-221], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.79999999999999997e-147 or 1.15e-221 < ew

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}, ew, \left(0 - eh\right) \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right)}\right| \]

   5. Taylor expanded in t around 0

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

   7. Simplified 50.4%

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

   if -4.79999999999999997e-147 < ew < 1.15e-221

   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. neg-fabs N/A

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

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{\left|\frac{\left(0 - eh\right) \cdot \left(\frac{\tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right) - ew \cdot \cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}\right|} \]

   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 81.1

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

   7. Simplified 81.1%

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

   8. Taylor expanded in t around 0

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

   10. Simplified 43.5%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.8 \cdot 10^{-147}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 1.15 \cdot 10^{-221}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.8% accurate, 287.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {\left(\frac{eh \cdot \tan t}{0 - ew}\right)}^{2}}}, ew, \left(0 - eh\right) \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right)\right)}\right| \]

5. Taylor expanded in t around 0

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

7. Simplified 44.8%

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

Reproduce

herbie shell --seed 2024196 
(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)))))))