Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $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. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

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

Alternative 3: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(\sin t \cdot eh\right) - \sqrt{\mathsf{fma}\left(\cos \left(2 \cdot t\right), 0.5, 0.5\right)} \cdot ew\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (sin (atan (/ (* (- eh) t) ew))) (* (sin t) eh))
   (* (sqrt (fma (cos (* 2.0 t)) 0.5 0.5)) ew))))

C

double code(double eh, double ew, double t) {
	return fabs(((sin(atan(((-eh * t) / ew))) * (sin(t) * eh)) - (sqrt(fma(cos((2.0 * t)), 0.5, 0.5)) * ew)));
}

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(sin(atan(Float64(Float64(Float64(-eh) * t) / ew))) * Float64(sin(t) * eh)) - Float64(sqrt(fma(cos(Float64(2.0 * t)), 0.5, 0.5)) * ew)))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(N[Cos[N[(2.0 * t), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] * ew), $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. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 99.1

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

5. Applied rewrites 99.1%

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

7. Applied rewrites 73.4%

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

8. Taylor expanded in ew around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lower-cos.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.6

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

10. Applied rewrites 98.6%

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

11. Final simplification 98.6%

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

Alternative 4: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \cos t \cdot ew\\ t_3 := \left|t\_2\right|\\ t_4 := \left(-eh\right) \cdot t\\ \mathbf{if}\;ew \leq -5.5 \cdot 10^{-76}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;ew \leq 6.8 \cdot 10^{-101}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;ew \leq 9.6 \cdot 10^{+92}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t\_4}{\sqrt{\frac{t\_4 \cdot t\_4}{ew \cdot ew} + 1} \cdot ew}, t\_1, -t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (* (cos t) ew))
        (t_3 (fabs t_2))
        (t_4 (* (- eh) t)))
   (if (<= ew -5.5e-76)
     t_3
     (if (<= ew 6.8e-101)
       (fabs t_1)
       (if (<= ew 9.6e+92)
         (fabs
          (fma
           (/ t_4 (* (sqrt (+ (/ (* t_4 t_4) (* ew ew)) 1.0)) ew))
           t_1
           (- t_2)))
         t_3)))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = cos(t) * ew;
	double t_3 = fabs(t_2);
	double t_4 = -eh * t;
	double tmp;
	if (ew <= -5.5e-76) {
		tmp = t_3;
	} else if (ew <= 6.8e-101) {
		tmp = fabs(t_1);
	} else if (ew <= 9.6e+92) {
		tmp = fabs(fma((t_4 / (sqrt((((t_4 * t_4) / (ew * ew)) + 1.0)) * ew)), t_1, -t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(cos(t) * ew)
	t_3 = abs(t_2)
	t_4 = Float64(Float64(-eh) * t)
	tmp = 0.0
	if (ew <= -5.5e-76)
		tmp = t_3;
	elseif (ew <= 6.8e-101)
		tmp = abs(t_1);
	elseif (ew <= 9.6e+92)
		tmp = abs(fma(Float64(t_4 / Float64(sqrt(Float64(Float64(Float64(t_4 * t_4) / Float64(ew * ew)) + 1.0)) * ew)), t_1, Float64(-t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[Abs[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[((-eh) * t), $MachinePrecision]}, If[LessEqual[ew, -5.5e-76], t$95$3, If[LessEqual[ew, 6.8e-101], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[ew, 9.6e+92], N[Abs[N[(N[(t$95$4 / N[(N[Sqrt[N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] / N[(ew * ew), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * t$95$1 + (-t$95$2)), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -5.50000000000000014e-76 or 9.60000000000000018e92 < ew

   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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if -5.50000000000000014e-76 < ew < 6.79999999999999978e-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

   4. Applied rewrites 37.0%

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

   5. Taylor expanded in eh around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 86.7

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

   7. Applied rewrites 86.7%

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

   if 6.79999999999999978e-101 < ew < 9.60000000000000018e92

   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. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.3

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

   10. Applied rewrites 98.3%

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

   12. Applied rewrites 77.9%

       \[\leadsto \color{blue}{\left|\mathsf{fma}\left(\frac{\left(-eh\right) \cdot t}{\sqrt{\frac{\left(\left(-eh\right) \cdot t\right) \cdot \left(\left(-eh\right) \cdot t\right)}{ew \cdot ew} + 1} \cdot ew}, \sin t \cdot eh, -\cos t \cdot ew\right)\right|} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos t \cdot ew\right|\\ \mathbf{if}\;ew \leq -5.5 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.5 \cdot 10^{-84}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (cos t) ew))))
   (if (<= ew -5.5e-76) t_1 (if (<= ew 3.5e-84) (fabs (* (sin t) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(t) * ew));
	double tmp;
	if (ew <= -5.5e-76) {
		tmp = t_1;
	} else if (ew <= 3.5e-84) {
		tmp = fabs((sin(t) * eh));
	} 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 <= (-5.5d-76)) then
        tmp = t_1
    else if (ew <= 3.5d-84) then
        tmp = abs((sin(t) * eh))
    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 <= -5.5e-76) {
		tmp = t_1;
	} else if (ew <= 3.5e-84) {
		tmp = Math.abs((Math.sin(t) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(t) * ew))
	tmp = 0
	if ew <= -5.5e-76:
		tmp = t_1
	elif ew <= 3.5e-84:
		tmp = math.fabs((math.sin(t) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(t) * ew))
	tmp = 0.0
	if (ew <= -5.5e-76)
		tmp = t_1;
	elseif (ew <= 3.5e-84)
		tmp = abs(Float64(sin(t) * eh));
	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 <= -5.5e-76)
		tmp = t_1;
	elseif (ew <= 3.5e-84)
		tmp = abs((sin(t) * eh));
	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, -5.5e-76], t$95$1, If[LessEqual[ew, 3.5e-84], N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.50000000000000014e-76 or 3.5000000000000001e-84 < ew

   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

   4. Applied rewrites 46.5%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if -5.50000000000000014e-76 < ew < 3.5000000000000001e-84

   1. Initial program 99.7%

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

   4. Applied rewrites 38.9%

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

   5. Taylor expanded in eh around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 86.1

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

   7. Applied rewrites 86.1%

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

Alternative 6: 63.1% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 36.3%

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

5. Taylor expanded in ew around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 56.3

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

7. Applied rewrites 56.3%

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

Alternative 7: 43.1% 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

4. Applied rewrites 70.9%

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

5. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 36.1

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

7. Applied rewrites 36.1%

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

9. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2024240 
(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)))))))