Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.9s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t))))
   (fabs
    (+
     (* ew (/ (sin t) (hypot 1.0 t_1)))
     (* (* eh (cos t)) (sin (atan t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	return fabs(((ew * (sin(t) / hypot(1.0, t_1))) + ((eh * cos(t)) * sin(atan(t_1)))));
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / Math.tan(t);
	return Math.abs(((ew * (Math.sin(t) / Math.hypot(1.0, t_1))) + ((eh * Math.cos(t)) * Math.sin(Math.atan(t_1)))));
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / math.tan(t)
	return math.fabs(((ew * (math.sin(t) / math.hypot(1.0, t_1))) + ((eh * math.cos(t)) * math.sin(math.atan(t_1)))))

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(Float64(Float64(ew * Float64(sin(t) / hypot(1.0, t_1))) + Float64(Float64(eh * cos(t)) * sin(atan(t_1)))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / tan(t);
	tmp = abs(((ew * (sin(t) / hypot(1.0, t_1))) + ((eh * cos(t)) * sin(atan(t_1)))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(ew * N[(N[Sin[t], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   4. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin t, \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   8. hypot-1-def N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   12. tan-lowering-tan.f64 99.8%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot t\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(t \cdot ew\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 99.0%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right), \mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(t, ew\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   4. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin t, \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   8. hypot-1-def N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   12. tan-lowering-tan.f64 99.8%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eh around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\sin t}, ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. sin-lowering-sin.f64 97.8%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 4: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{if}\;eh \leq -5.1 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+76}:\\ \;\;\;\;\left|\sin t \cdot ew + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* eh (cos t)) (sin (atan (/ eh (* ew (tan t)))))))))
   (if (<= eh -5.1e+158)
     t_1
     (if (<= eh 8.5e+76)
       (fabs (+ (* (sin t) ew) (* eh (sin (atan (/ (/ eh ew) (tan t)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	double tmp;
	if (eh <= -5.1e+158) {
		tmp = t_1;
	} else if (eh <= 8.5e+76) {
		tmp = fabs(((sin(t) * ew) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))))
    if (eh <= (-5.1d+158)) then
        tmp = t_1
    else if (eh <= 8.5d+76) then
        tmp = abs(((sin(t) * ew) + (eh * sin(atan(((eh / ew) / tan(t)))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	double tmp;
	if (eh <= -5.1e+158) {
		tmp = t_1;
	} else if (eh <= 8.5e+76) {
		tmp = Math.abs(((Math.sin(t) * ew) + (eh * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((eh * math.cos(t)) * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	tmp = 0
	if eh <= -5.1e+158:
		tmp = t_1
	elif eh <= 8.5e+76:
		tmp = math.fabs(((math.sin(t) * ew) + (eh * math.sin(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))
	tmp = 0.0
	if (eh <= -5.1e+158)
		tmp = t_1;
	elseif (eh <= 8.5e+76)
		tmp = abs(Float64(Float64(sin(t) * ew) + Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	tmp = 0.0;
	if (eh <= -5.1e+158)
		tmp = t_1;
	elseif (eh <= 8.5e+76)
		tmp = abs(((sin(t) * ew) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -5.1e+158], t$95$1, If[LessEqual[eh, 8.5e+76], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.09999999999999987e158 or 8.49999999999999992e76 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \cos t\right)\right)\right) \]

      10. cos-lowering-cos.f64 92.9%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right)\right)\right) \]

   5. Simplified 92.9%

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

   if -5.09999999999999987e158 < eh < 8.49999999999999992e76

   1. Initial program 99.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      4. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin t, \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      8. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      12. tan-lowering-tan.f64 99.9%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\color{blue}{eh}, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 90.7%

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

   8. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\sin t}, ew\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. sin-lowering-sin.f64 89.3%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), ew\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 89.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.1 \cdot 10^{+158}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+76}:\\ \;\;\;\;\left|\sin t \cdot ew + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{if}\;eh \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 9 \cdot 10^{-61}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* eh (cos t)) (sin (atan (/ eh (* ew (tan t)))))))))
   (if (<= eh -2.5e-59) t_1 (if (<= eh 9e-61) (fabs (* (sin t) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	double tmp;
	if (eh <= -2.5e-59) {
		tmp = t_1;
	} else if (eh <= 9e-61) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))))
    if (eh <= (-2.5d-59)) then
        tmp = t_1
    else if (eh <= 9d-61) then
        tmp = abs((sin(t) * ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	double tmp;
	if (eh <= -2.5e-59) {
		tmp = t_1;
	} else if (eh <= 9e-61) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((eh * math.cos(t)) * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	tmp = 0
	if eh <= -2.5e-59:
		tmp = t_1
	elif eh <= 9e-61:
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))
	tmp = 0.0
	if (eh <= -2.5e-59)
		tmp = t_1;
	elseif (eh <= 9e-61)
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	tmp = 0.0;
	if (eh <= -2.5e-59)
		tmp = t_1;
	elseif (eh <= 9e-61)
		tmp = abs((sin(t) * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.5e-59], t$95$1, If[LessEqual[eh, 9e-61], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.5000000000000001e-59 or 9e-61 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \left(eh \cdot \cos t\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \cos t\right)\right)\right) \]

      10. cos-lowering-cos.f64 82.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right)\right)\right) \]

   5. Simplified 82.0%

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

   if -2.5000000000000001e-59 < eh < 9e-61

   1. Initial program 99.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      4. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin t, \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      8. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      12. tan-lowering-tan.f64 99.9%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 71.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 71.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 9 \cdot 10^{-61}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 8.2 \cdot 10^{-60}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot \left(eh \cdot \left(0.5 - 0.5 \cdot \cos \left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -2e-59)
   (fabs eh)
   (if (<= eh 8.2e-60)
     (fabs (* (sin t) ew))
     (*
      (cos t)
      (* eh (- 0.5 (* 0.5 (cos (* (atan (/ eh (* ew (tan t)))) 2.0)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2e-59) {
		tmp = fabs(eh);
	} else if (eh <= 8.2e-60) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = cos(t) * (eh * (0.5 - (0.5 * cos((atan((eh / (ew * tan(t)))) * 2.0)))));
	}
	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 <= (-2d-59)) then
        tmp = abs(eh)
    else if (eh <= 8.2d-60) then
        tmp = abs((sin(t) * ew))
    else
        tmp = cos(t) * (eh * (0.5d0 - (0.5d0 * cos((atan((eh / (ew * tan(t)))) * 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2e-59) {
		tmp = Math.abs(eh);
	} else if (eh <= 8.2e-60) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = Math.cos(t) * (eh * (0.5 - (0.5 * Math.cos((Math.atan((eh / (ew * Math.tan(t)))) * 2.0)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -2e-59:
		tmp = math.fabs(eh)
	elif eh <= 8.2e-60:
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = math.cos(t) * (eh * (0.5 - (0.5 * math.cos((math.atan((eh / (ew * math.tan(t)))) * 2.0)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -2e-59)
		tmp = abs(eh);
	elseif (eh <= 8.2e-60)
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = Float64(cos(t) * Float64(eh * Float64(0.5 - Float64(0.5 * cos(Float64(atan(Float64(eh / Float64(ew * tan(t)))) * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -2e-59)
		tmp = abs(eh);
	elseif (eh <= 8.2e-60)
		tmp = abs((sin(t) * ew));
	else
		tmp = cos(t) * (eh * (0.5 - (0.5 * cos((atan((eh / (ew * tan(t)))) * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -2e-59], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 8.2e-60], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * N[(eh * N[(0.5 - N[(0.5 * N[Cos[N[(N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -2.0000000000000001e-59

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right)\right)\right) \]

      6. tan-lowering-tan.f64 49.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 49.4%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{eh}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(\left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right), \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 30.9%

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

   8. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{eh}\right) \]
   9. Step-by-step derivation

   10. Simplified 49.7%

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

   if -2.0000000000000001e-59 < eh < 8.20000000000000025e-60

   1. Initial program 99.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      4. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin t, \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      8. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      12. tan-lowering-tan.f64 99.9%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 71.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 71.4%

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

   if 8.20000000000000025e-60 < eh

   1. Initial program 99.8%

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

   4. Applied egg-rr 30.6%

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

   5. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{eh \cdot \left(\cos t \cdot {\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}^{2}\right)}\right)}\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(eh \cdot \left(\cos t \cdot {\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}^{2}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\left(eh \cdot \cos t\right) \cdot {\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(eh \cdot \cos t\right), \left({\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \cos t\right), \left({\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \left({\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}^{2}\right)\right)\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{pow.f64}\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right), 2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right), 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right), 2\right)\right)\right)\right)\right) \]

      11. tan-lowering-tan.f64 80.8%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), 2\right)\right)\right)\right)\right) \]

   7. Simplified 80.8%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. fabs-sqr N/A

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

      4. sqr-pow N/A

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

      5. inv-pow N/A

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

      6. remove-double-div N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}^{2} \cdot eh\right), \color{blue}{\cos t}\right) \]

   9. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \cdot eh\right) \cdot \cos t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 8.2 \cdot 10^{-60}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot \left(eh \cdot \left(0.5 - 0.5 \cdot \cos \left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{-41}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{1}{t\_1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)))
   (if (<= t -4.1e-41)
     (fabs t_1)
     (if (<= t 1.75e-7) (fabs eh) (fabs (/ 1.0 (/ 1.0 t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double tmp;
	if (t <= -4.1e-41) {
		tmp = fabs(t_1);
	} else if (t <= 1.75e-7) {
		tmp = fabs(eh);
	} else {
		tmp = fabs((1.0 / (1.0 / 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 = sin(t) * ew
    if (t <= (-4.1d-41)) then
        tmp = abs(t_1)
    else if (t <= 1.75d-7) then
        tmp = abs(eh)
    else
        tmp = abs((1.0d0 / (1.0d0 / t_1)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * ew;
	double tmp;
	if (t <= -4.1e-41) {
		tmp = Math.abs(t_1);
	} else if (t <= 1.75e-7) {
		tmp = Math.abs(eh);
	} else {
		tmp = Math.abs((1.0 / (1.0 / t_1)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * ew
	tmp = 0
	if t <= -4.1e-41:
		tmp = math.fabs(t_1)
	elif t <= 1.75e-7:
		tmp = math.fabs(eh)
	else:
		tmp = math.fabs((1.0 / (1.0 / t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	tmp = 0.0
	if (t <= -4.1e-41)
		tmp = abs(t_1);
	elseif (t <= 1.75e-7)
		tmp = abs(eh);
	else
		tmp = abs(Float64(1.0 / Float64(1.0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * ew;
	tmp = 0.0;
	if (t <= -4.1e-41)
		tmp = abs(t_1);
	elseif (t <= 1.75e-7)
		tmp = abs(eh);
	else
		tmp = abs((1.0 / (1.0 / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[t, -4.1e-41], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[t, 1.75e-7], N[Abs[eh], $MachinePrecision], N[Abs[N[(1.0 / N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.10000000000000014e-41

   1. Initial program 99.7%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      4. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin t, \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      8. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      12. tan-lowering-tan.f64 99.8%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 51.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 51.9%

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

   if -4.10000000000000014e-41 < t < 1.74999999999999992e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right)\right)\right) \]

      6. tan-lowering-tan.f64 74.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 74.5%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{eh}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(\left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right), \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 42.8%

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

   8. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{eh}\right) \]
   9. Step-by-step derivation

   10. Simplified 74.9%

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

   if 1.74999999999999992e-7 < t

   1. Initial program 99.6%

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

   4. Applied egg-rr 38.9%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{ew \cdot \sin t}\right)}\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(ew \cdot \sin t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(ew, \sin t\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 50.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right) \]

   7. Simplified 50.9%

      \[\leadsto \left|\frac{1}{\color{blue}{\frac{1}{ew \cdot \sin t}}}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-41}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{1}{\sin t \cdot ew}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -1.35e-41) t_1 (if (<= t 2.3e-7) (fabs eh) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -1.35e-41) {
		tmp = t_1;
	} else if (t <= 2.3e-7) {
		tmp = fabs(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((sin(t) * ew))
    if (t <= (-1.35d-41)) then
        tmp = t_1
    else if (t <= 2.3d-7) then
        tmp = abs(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.sin(t) * ew));
	double tmp;
	if (t <= -1.35e-41) {
		tmp = t_1;
	} else if (t <= 2.3e-7) {
		tmp = Math.abs(eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if t <= -1.35e-41:
		tmp = t_1
	elif t <= 2.3e-7:
		tmp = math.fabs(eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -1.35e-41)
		tmp = t_1;
	elseif (t <= 2.3e-7)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (t <= -1.35e-41)
		tmp = t_1;
	elseif (t <= 2.3e-7)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.35e-41], t$95$1, If[LessEqual[t, 2.3e-7], N[Abs[eh], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e-41 or 2.29999999999999995e-7 < t

   1. Initial program 99.7%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      4. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin t, \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      8. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      12. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 51.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 51.5%

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

   if -1.35e-41 < t < 2.29999999999999995e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right)\right)\right) \]

      6. tan-lowering-tan.f64 74.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 74.5%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{eh}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(\left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right), \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 42.8%

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

   8. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{eh}\right) \]
   9. Step-by-step derivation

   10. Simplified 74.9%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-41}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 2.5e+99) (fabs eh) (fabs (* t ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 2.5e+99) {
		tmp = fabs(eh);
	} else {
		tmp = fabs((t * ew));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 2.5d+99) then
        tmp = abs(eh)
    else
        tmp = abs((t * ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 2.5e+99) {
		tmp = Math.abs(eh);
	} else {
		tmp = Math.abs((t * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 2.5e+99:
		tmp = math.fabs(eh)
	else:
		tmp = math.fabs((t * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 2.5e+99)
		tmp = abs(eh);
	else
		tmp = abs(Float64(t * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 2.5e+99)
		tmp = abs(eh);
	else
		tmp = abs((t * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 2.5e+99], N[Abs[eh], $MachinePrecision], N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 2.50000000000000004e99

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right)\right)\right) \]

      6. tan-lowering-tan.f64 48.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 48.4%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

      2. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{eh}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(\left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right), \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 28.1%

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

   8. Taylor expanded in eh around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{eh}\right) \]
   9. Step-by-step derivation

   10. Simplified 48.8%

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

   if 2.50000000000000004e99 < ew

   1. Initial program 99.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      4. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sin t \cdot \frac{1}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin t}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin t, \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      8. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \left(\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{eh}{ew}\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \tan t\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

      12. tan-lowering-tan.f64 99.8%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(t\right), \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right), ew\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(eh, ew\right), \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 81.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 81.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot t\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(t \cdot ew\right)\right) \]

      2. *-lowering-*.f64 39.7%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, ew\right)\right) \]

   10. Simplified 39.7%

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

Alternative 10: 42.3% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \left(ew \cdot \tan t\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \tan t\right)\right)\right)\right)\right)\right) \]

   6. tan-lowering-tan.f64 43.5%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 43.5%

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

   1. associate-/r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right) \]

   2. sin-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{\frac{eh}{ew}}{\tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{\frac{eh}{ew \cdot \tan t}}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}}\right)\right)\right) \]

   5. associate-/l/ N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \left(\frac{eh}{\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}} \cdot \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(\left(\sqrt{1 \cdot 1 + \frac{\frac{eh}{ew}}{\tan t} \cdot \frac{\frac{eh}{ew}}{\tan t}}\right), \left(ew \cdot \tan t\right)\right)\right)\right)\right) \]

7. Applied egg-rr 26.4%

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

8. Taylor expanded in eh around inf

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{eh}\right) \]
9. Step-by-step derivation

10. Simplified 43.9%

    \[\leadsto \left|\color{blue}{eh}\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))