Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.3s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((tan(t) * eh) / -ew));
	tmp = abs(((cos(t_1) * (cos(t) * ew)) - (sin(t_1) * (sin(t) * eh))));
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[Cos[t$95$1], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $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%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((sin(atan(((tan(t) * eh) / -ew))) * (sin(t) * eh)) - ((cos(t) / sqrt((pow(((tan(t) / ew) * eh), 2.0) + 1.0))) * 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(((tan(t) * eh) / -ew))) * (sin(t) * eh)) - ((cos(t) / sqrt(((((tan(t) / ew) * eh) ** 2.0d0) + 1.0d0))) * ew)))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((Math.sin(Math.atan(((Math.tan(t) * eh) / -ew))) * (Math.sin(t) * eh)) - ((Math.cos(t) / Math.sqrt((Math.pow(((Math.tan(t) / ew) * eh), 2.0) + 1.0))) * ew)));
}

Python

def code(eh, ew, t):
	return math.fabs(((math.sin(math.atan(((math.tan(t) * eh) / -ew))) * (math.sin(t) * eh)) - ((math.cos(t) / math.sqrt((math.pow(((math.tan(t) / ew) * eh), 2.0) + 1.0))) * ew)))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((sin(atan(((tan(t) * eh) / -ew))) * (sin(t) * eh)) - ((cos(t) / sqrt(((((tan(t) / ew) * eh) ^ 2.0) + 1.0))) * ew)));
end

Wolfram

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 98.1% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (/ (cos t) (/ 1.0 ew))
   (* (sin (atan (/ (* (tan t) eh) (- ew)))) (* (sin t) eh)))))

C

double code(double eh, double ew, double t) {
	return fabs(((cos(t) / (1.0 / ew)) - (sin(atan(((tan(t) * eh) / -ew))) * (sin(t) * eh))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((cos(t) / (1.0d0 / ew)) - (sin(atan(((tan(t) * eh) / -ew))) * (sin(t) * eh))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((Math.cos(t) / (1.0 / ew)) - (Math.sin(Math.atan(((Math.tan(t) * eh) / -ew))) * (Math.sin(t) * eh))));
}

Python

def code(eh, ew, t):
	return math.fabs(((math.cos(t) / (1.0 / ew)) - (math.sin(math.atan(((math.tan(t) * eh) / -ew))) * (math.sin(t) * eh))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(cos(t) / Float64(1.0 / ew)) - Float64(sin(atan(Float64(Float64(tan(t) * eh) / Float64(-ew)))) * Float64(sin(t) * eh))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((cos(t) / (1.0 / ew)) - (sin(atan(((tan(t) * eh) / -ew))) * (sin(t) * eh))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

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

Alternative 4: 74.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{\mathsf{fma}\left(\left(0.5 \cdot eh\right) \cdot \left(\frac{\sin t}{ew} \cdot \tan t\right), eh, \cos t \cdot ew\right)}\right|}\\ \mathbf{if}\;t \leq -0.00085:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.00047:\\ \;\;\;\;\left|ew - \left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (/
          1.0
          (fabs
           (/
            1.0
            (fma
             (* (* 0.5 eh) (* (/ (sin t) ew) (tan t)))
             eh
             (* (cos t) ew)))))))
   (if (<= t -0.00085)
     t_1
     (if (<= t 0.00047)
       (fabs (- ew (* (* eh t) (sin (atan (/ (* (tan t) eh) (- ew)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / fma(((0.5 * eh) * ((sin(t) / ew) * tan(t))), eh, (cos(t) * ew))));
	double tmp;
	if (t <= -0.00085) {
		tmp = t_1;
	} else if (t <= 0.00047) {
		tmp = fabs((ew - ((eh * t) * sin(atan(((tan(t) * eh) / -ew))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / fma(Float64(Float64(0.5 * eh) * Float64(Float64(sin(t) / ew) * tan(t))), eh, Float64(cos(t) * ew)))))
	tmp = 0.0
	if (t <= -0.00085)
		tmp = t_1;
	elseif (t <= 0.00047)
		tmp = abs(Float64(ew - Float64(Float64(eh * t) * sin(atan(Float64(Float64(tan(t) * eh) / Float64(-ew)))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / N[(N[(N[(0.5 * eh), $MachinePrecision] * N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.00085], t$95$1, If[LessEqual[t, 0.00047], N[Abs[N[(ew - N[(N[(eh * t), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.49999999999999953e-4 or 4.69999999999999986e-4 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 50.2%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\mathsf{fma}\left(0.5 \cdot \frac{{\sin t}^{2}}{\cos t \cdot ew}, eh \cdot eh, \cos t \cdot ew\right)}}\right|} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.8%

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

   if -8.49999999999999953e-4 < t < 4.69999999999999986e-4

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-tan.f64 98.9

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

   7. Applied rewrites 98.9%

      \[\leadsto \left|\color{blue}{ew - \left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(-\frac{\tan t \cdot eh}{ew}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00085:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\mathsf{fma}\left(\left(0.5 \cdot eh\right) \cdot \left(\frac{\sin t}{ew} \cdot \tan t\right), eh, \cos t \cdot ew\right)}\right|}\\ \mathbf{elif}\;t \leq 0.00047:\\ \;\;\;\;\left|ew - \left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\mathsf{fma}\left(\left(0.5 \cdot eh\right) \cdot \left(\frac{\sin t}{ew} \cdot \tan t\right), eh, \cos t \cdot ew\right)}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{if}\;ew \leq -1.55 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{eh \cdot t}\right|}\\ \mathbf{elif}\;ew \leq 2.3 \cdot 10^{-176}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (fabs (/ 1.0 ew)))))
   (if (<= ew -1.55e-25)
     t_1
     (if (<= ew 1.3e-307)
       (/ 1.0 (fabs (/ 1.0 (* eh t))))
       (if (<= ew 2.3e-176) (* (sin t) eh) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / ew));
	double tmp;
	if (ew <= -1.55e-25) {
		tmp = t_1;
	} else if (ew <= 1.3e-307) {
		tmp = 1.0 / fabs((1.0 / (eh * t)));
	} else if (ew <= 2.3e-176) {
		tmp = 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 = 1.0d0 / abs((1.0d0 / ew))
    if (ew <= (-1.55d-25)) then
        tmp = t_1
    else if (ew <= 1.3d-307) then
        tmp = 1.0d0 / abs((1.0d0 / (eh * t)))
    else if (ew <= 2.3d-176) then
        tmp = 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 = 1.0 / Math.abs((1.0 / ew));
	double tmp;
	if (ew <= -1.55e-25) {
		tmp = t_1;
	} else if (ew <= 1.3e-307) {
		tmp = 1.0 / Math.abs((1.0 / (eh * t)));
	} else if (ew <= 2.3e-176) {
		tmp = Math.sin(t) * eh;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = 1.0 / math.fabs((1.0 / ew))
	tmp = 0
	if ew <= -1.55e-25:
		tmp = t_1
	elif ew <= 1.3e-307:
		tmp = 1.0 / math.fabs((1.0 / (eh * t)))
	elif ew <= 2.3e-176:
		tmp = math.sin(t) * eh
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / ew)))
	tmp = 0.0
	if (ew <= -1.55e-25)
		tmp = t_1;
	elseif (ew <= 1.3e-307)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(eh * t))));
	elseif (ew <= 2.3e-176)
		tmp = Float64(sin(t) * eh);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = 1.0 / abs((1.0 / ew));
	tmp = 0.0;
	if (ew <= -1.55e-25)
		tmp = t_1;
	elseif (ew <= 1.3e-307)
		tmp = 1.0 / abs((1.0 / (eh * t)));
	elseif (ew <= 2.3e-176)
		tmp = sin(t) * eh;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.55e-25], t$95$1, If[LessEqual[ew, 1.3e-307], N[(1.0 / N[Abs[N[(1.0 / N[(eh * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 2.3e-176], N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.54999999999999997e-25 or 2.3000000000000001e-176 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 52.0

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

   7. Applied rewrites 52.0%

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

   if -1.54999999999999997e-25 < ew < 1.29999999999999998e-307

   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 56.0%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      3. lower-sin.f64 70.9

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot eh}\right|} \]

   7. Applied rewrites 70.9%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 41.8%

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

   if 1.29999999999999998e-307 < ew < 2.3000000000000001e-176

   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 43.1%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      3. lower-sin.f64 81.9

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot eh}\right|} \]

   7. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left|\frac{1}{\sin t \cdot eh}\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left|\frac{1}{\sin t \cdot eh}\right|}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{\sin t \cdot eh}}\right|} \]

      4. inv-pow N/A

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

   9. Applied rewrites 52.1%

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

Alternative 6: 74.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos t \cdot ew\right|\\ \mathbf{if}\;ew \leq -3.35 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (cos t) ew))))
   (if (<= ew -3.35e-25) t_1 (if (<= ew 3.5e-115) (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 <= -3.35e-25) {
		tmp = t_1;
	} else if (ew <= 3.5e-115) {
		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 <= (-3.35d-25)) then
        tmp = t_1
    else if (ew <= 3.5d-115) 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 <= -3.35e-25) {
		tmp = t_1;
	} else if (ew <= 3.5e-115) {
		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 <= -3.35e-25:
		tmp = t_1
	elif ew <= 3.5e-115:
		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 <= -3.35e-25)
		tmp = t_1;
	elseif (ew <= 3.5e-115)
		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 <= -3.35e-25)
		tmp = t_1;
	elseif (ew <= 3.5e-115)
		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, -3.35e-25], t$95$1, If[LessEqual[ew, 3.5e-115], N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.35000000000000016e-25 or 3.5000000000000002e-115 < 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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   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 79.9

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

   7. Applied rewrites 79.9%

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

   if -3.35000000000000016e-25 < ew < 3.5000000000000002e-115

   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 55.0%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      3. lower-sin.f64 72.1

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot eh}\right|} \]

   7. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left|\frac{1}{\sin t \cdot eh}\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left|\frac{1}{\sin t \cdot eh}\right|}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{\sin t \cdot eh}}\right|} \]

      4. fabs-div N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left|1\right|}{\left|\sin t \cdot eh\right|}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left|\sin t \cdot eh\right|}} \]

   9. Applied rewrites 72.2%

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

Alternative 7: 62.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 1.6 \cdot 10^{+195}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{eh \cdot t}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 1.6e+195) (fabs (* (cos t) ew)) (/ 1.0 (fabs (/ 1.0 (* eh t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.6e+195) {
		tmp = fabs((cos(t) * ew));
	} else {
		tmp = 1.0 / fabs((1.0 / (eh * t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= 1.6d+195) then
        tmp = abs((cos(t) * ew))
    else
        tmp = 1.0d0 / abs((1.0d0 / (eh * t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.6e+195) {
		tmp = Math.abs((Math.cos(t) * ew));
	} else {
		tmp = 1.0 / Math.abs((1.0 / (eh * t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 1.6e+195:
		tmp = math.fabs((math.cos(t) * ew))
	else:
		tmp = 1.0 / math.fabs((1.0 / (eh * t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 1.6e+195)
		tmp = abs(Float64(cos(t) * ew));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(eh * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 1.6e+195)
		tmp = abs((cos(t) * ew));
	else
		tmp = 1.0 / abs((1.0 / (eh * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 1.6e+195], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / N[(eh * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.59999999999999991e195

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   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 67.1

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

   7. Applied rewrites 67.1%

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

   if 1.59999999999999991e195 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 28.5%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      3. lower-sin.f64 89.0

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot eh}\right|} \]

   7. Applied rewrites 89.0%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 49.9%

       \[\leadsto \frac{1}{\left|\frac{1}{eh \cdot \color{blue}{t}}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 43.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{if}\;ew \leq -1.55 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{eh \cdot t}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (fabs (/ 1.0 ew)))))
   (if (<= ew -1.55e-25)
     t_1
     (if (<= ew 1.1e-161) (/ 1.0 (fabs (/ 1.0 (* eh t)))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / ew));
	double tmp;
	if (ew <= -1.55e-25) {
		tmp = t_1;
	} else if (ew <= 1.1e-161) {
		tmp = 1.0 / fabs((1.0 / (eh * 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 = 1.0d0 / abs((1.0d0 / ew))
    if (ew <= (-1.55d-25)) then
        tmp = t_1
    else if (ew <= 1.1d-161) then
        tmp = 1.0d0 / abs((1.0d0 / (eh * 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 = 1.0 / Math.abs((1.0 / ew));
	double tmp;
	if (ew <= -1.55e-25) {
		tmp = t_1;
	} else if (ew <= 1.1e-161) {
		tmp = 1.0 / Math.abs((1.0 / (eh * t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = 1.0 / math.fabs((1.0 / ew))
	tmp = 0
	if ew <= -1.55e-25:
		tmp = t_1
	elif ew <= 1.1e-161:
		tmp = 1.0 / math.fabs((1.0 / (eh * t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / ew)))
	tmp = 0.0
	if (ew <= -1.55e-25)
		tmp = t_1;
	elseif (ew <= 1.1e-161)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(eh * t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = 1.0 / abs((1.0 / ew));
	tmp = 0.0;
	if (ew <= -1.55e-25)
		tmp = t_1;
	elseif (ew <= 1.1e-161)
		tmp = 1.0 / abs((1.0 / (eh * t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.55e-25], t$95$1, If[LessEqual[ew, 1.1e-161], N[(1.0 / N[Abs[N[(1.0 / N[(eh * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.54999999999999997e-25 or 1.10000000000000001e-161 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 53.0

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

   7. Applied rewrites 53.0%

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

   if -1.54999999999999997e-25 < ew < 1.10000000000000001e-161

   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 55.2%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

      3. lower-sin.f64 72.9

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot eh}\right|} \]

   7. Applied rewrites 72.9%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot eh}}\right|} \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 37.4%

       \[\leadsto \frac{1}{\left|\frac{1}{eh \cdot \color{blue}{t}}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 41.7% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left|\frac{1}{ew}\right|} \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (/ 1.0 (fabs (/ 1.0 ew))))

C

double code(double eh, double ew, double t) {
	return 1.0 / fabs((1.0 / 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 = 1.0d0 / abs((1.0d0 / ew))
end function

Java

public static double code(double eh, double ew, double t) {
	return 1.0 / Math.abs((1.0 / ew));
}

Python

def code(eh, ew, t):
	return 1.0 / math.fabs((1.0 / ew))

Julia

function code(eh, ew, t)
	return Float64(1.0 / abs(Float64(1.0 / ew)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = 1.0 / abs((1.0 / ew));
end

Wolfram

code[eh_, ew_, t_] := N[(1.0 / N[Abs[N[(1.0 / 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

4. Applied rewrites 77.4%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 41.5

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

7. Applied rewrites 41.5%

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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)))))))