Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.0s

Alternatives: 11

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(fma(Float64(cos(t) * sin(atan(t_1))), eh, Float64(Float64(ew * sin(t)) / hypot(1.0, 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[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $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. +-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. fma-define N/A

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

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

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

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)) + (sin(atan(t_1)) * (cos(t) * eh))));
}

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)) + (Math.sin(Math.atan(t_1)) * (Math.cos(t) * eh))));
}

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)) + (math.sin(math.atan(t_1)) * (math.cos(t) * eh))))

Julia

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

MATLAB

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

Wolfram

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

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

   6. hypot-1-def N/A

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

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

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

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

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

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

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

   10. tan-lowering-tan.f64 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / tan(t);
	tmp = abs(((sin(atan(t_1)) * (cos(t) * eh)) + (ew * (sin(t) / hypot(1.0, 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[(N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] + N[(ew * N[(N[Sin[t], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $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%

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

Alternative 4: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision] ^ 2], $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. 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 99.5%

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

5. Simplified 99.5%

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

   1. *-commutative N/A

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

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

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

   3. cos-atan N/A

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

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

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

   5. hypot-1-def N/A

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

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

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. fma-define N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eh around 0

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

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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 6: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ \mathbf{if}\;eh \leq -8.5 \cdot 10^{-89}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)))
   (if (<= eh -8.5e-89)
     (fabs (* t_1 (sin (atan (/ eh (* ew (tan t)))))))
     (if (<= eh 5e-56)
       (fabs (* ew (sin t)))
       (fabs (* (sin (atan (/ (/ eh ew) (tan t)))) t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double tmp;
	if (eh <= -8.5e-89) {
		tmp = fabs((t_1 * sin(atan((eh / (ew * tan(t)))))));
	} else if (eh <= 5e-56) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((sin(atan(((eh / ew) / tan(t)))) * 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 = cos(t) * eh
    if (eh <= (-8.5d-89)) then
        tmp = abs((t_1 * sin(atan((eh / (ew * tan(t)))))))
    else if (eh <= 5d-56) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs((sin(atan(((eh / ew) / tan(t)))) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.cos(t) * eh;
	double tmp;
	if (eh <= -8.5e-89) {
		tmp = Math.abs((t_1 * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	} else if (eh <= 5e-56) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((eh / ew) / Math.tan(t)))) * t_1));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * eh
	tmp = 0
	if eh <= -8.5e-89:
		tmp = math.fabs((t_1 * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	elif eh <= 5e-56:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((math.sin(math.atan(((eh / ew) / math.tan(t)))) * t_1))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	tmp = 0.0
	if (eh <= -8.5e-89)
		tmp = abs(Float64(t_1 * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	elseif (eh <= 5e-56)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / ew) / tan(t)))) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * eh;
	tmp = 0.0;
	if (eh <= -8.5e-89)
		tmp = abs((t_1 * sin(atan((eh / (ew * tan(t)))))));
	elseif (eh <= 5e-56)
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((sin(atan(((eh / ew) / tan(t)))) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -8.5e-89], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 5e-56], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -8.49999999999999937e-89

   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 80.7%

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

   5. Simplified 80.7%

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

   if -8.49999999999999937e-89 < eh < 4.99999999999999997e-56

   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. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   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 74.8%

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

   7. Simplified 74.8%

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

   if 4.99999999999999997e-56 < eh

   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. Step-by-step derivation

      1. cos-atan N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. hypot-1-def N/A

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

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

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

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

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

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

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

      10. tan-lowering-tan.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right) \]
   6. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      7. associate-/r* N/A

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

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

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

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

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

      10. tan-lowering-tan.f64 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.5 \cdot 10^{-89}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{if}\;eh \leq -1.8 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 8.8 \cdot 10^{-57}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* (cos t) eh) (sin (atan (/ eh (* ew (tan t)))))))))
   (if (<= eh -1.8e-88) t_1 (if (<= eh 8.8e-57) (fabs (* ew (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((cos(t) * eh) * sin(atan((eh / (ew * tan(t)))))));
	double tmp;
	if (eh <= -1.8e-88) {
		tmp = t_1;
	} else if (eh <= 8.8e-57) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(((cos(t) * eh) * sin(atan((eh / (ew * tan(t)))))))
    if (eh <= (-1.8d-88)) then
        tmp = t_1
    else if (eh <= 8.8d-57) then
        tmp = abs((ew * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(((Math.cos(t) * eh) * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	double tmp;
	if (eh <= -1.8e-88) {
		tmp = t_1;
	} else if (eh <= 8.8e-57) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((math.cos(t) * eh) * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	tmp = 0
	if eh <= -1.8e-88:
		tmp = t_1
	elif eh <= 8.8e-57:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(cos(t) * eh) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))
	tmp = 0.0
	if (eh <= -1.8e-88)
		tmp = t_1;
	elseif (eh <= 8.8e-57)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((cos(t) * eh) * sin(atan((eh / (ew * tan(t)))))));
	tmp = 0.0;
	if (eh <= -1.8e-88)
		tmp = t_1;
	elseif (eh <= 8.8e-57)
		tmp = abs((ew * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.8e-88], t$95$1, If[LessEqual[eh, 8.8e-57], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.8e-88 or 8.79999999999999994e-57 < eh

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   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 83.0%

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

   5. Simplified 83.0%

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

   if -1.8e-88 < eh < 8.79999999999999994e-57

   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. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   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 74.8%

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

   7. Simplified 74.8%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.8 \cdot 10^{-88}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 8.8 \cdot 10^{-57}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -5.2 \cdot 10^{-87}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 3.25 \cdot 10^{-37}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -5.2e-87)
   (fabs eh)
   (if (<= eh 3.25e-37) (fabs (* ew (sin t))) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -5.2e-87) {
		tmp = fabs(eh);
	} else if (eh <= 3.25e-37) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(eh);
	}
	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 <= (-5.2d-87)) then
        tmp = abs(eh)
    else if (eh <= 3.25d-37) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -5.2e-87) {
		tmp = Math.abs(eh);
	} else if (eh <= 3.25e-37) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -5.2e-87:
		tmp = math.fabs(eh)
	elif eh <= 3.25e-37:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -5.2e-87)
		tmp = abs(eh);
	elseif (eh <= 3.25e-37)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -5.2e-87)
		tmp = abs(eh);
	elseif (eh <= 3.25e-37)
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -5.2e-87], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 3.25e-37], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.20000000000000005e-87 or 3.2500000000000001e-37 < eh

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   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 62.8%

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

   5. Simplified 62.8%

      \[\leadsto \left|\color{blue}{eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}\right| \]
   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. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. hypot-undefine N/A

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

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

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

      13. associate-/r* N/A

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

      14. /-lowering-/.f64 N/A

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

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

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

   7. Applied egg-rr 32.4%

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

   8. Taylor expanded in eh around inf

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

   10. Simplified 63.1%

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

   if -5.20000000000000005e-87 < eh < 3.2500000000000001e-37

   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. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   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 73.9%

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

   7. Simplified 73.9%

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

Alternative 9: 46.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.1 \cdot 10^{+190}:\\ \;\;\;\;\left|t \cdot ew\right|\\ \mathbf{elif}\;ew \leq 1.05 \cdot 10^{+204}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sin t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.1e+190)
   (fabs (* t ew))
   (if (<= ew 1.05e+204) (fabs eh) (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.1e+190) {
		tmp = fabs((t * ew));
	} else if (ew <= 1.05e+204) {
		tmp = fabs(eh);
	} else {
		tmp = ew * sin(t);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-1.1d+190)) then
        tmp = abs((t * ew))
    else if (ew <= 1.05d+204) then
        tmp = abs(eh)
    else
        tmp = ew * sin(t)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.1e+190) {
		tmp = Math.abs((t * ew));
	} else if (ew <= 1.05e+204) {
		tmp = Math.abs(eh);
	} else {
		tmp = ew * Math.sin(t);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.1e+190:
		tmp = math.fabs((t * ew))
	elif ew <= 1.05e+204:
		tmp = math.fabs(eh)
	else:
		tmp = ew * math.sin(t)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.1e+190)
		tmp = abs(Float64(t * ew));
	elseif (ew <= 1.05e+204)
		tmp = abs(eh);
	else
		tmp = Float64(ew * sin(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.1e+190)
		tmp = abs((t * ew));
	elseif (ew <= 1.05e+204)
		tmp = abs(eh);
	else
		tmp = ew * sin(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.1e+190], N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.05e+204], N[Abs[eh], $MachinePrecision], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.1e190

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   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 86.3%

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

   7. Simplified 86.3%

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

   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 57.6%

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

   10. Simplified 57.6%

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

   if -1.1e190 < ew < 1.05e204

   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 56.6%

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

   5. Simplified 56.6%

      \[\leadsto \left|\color{blue}{eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}\right| \]
   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. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. hypot-undefine N/A

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

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

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

      13. associate-/r* N/A

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

      14. /-lowering-/.f64 N/A

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

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

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

   7. Applied egg-rr 28.5%

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

   8. Taylor expanded in eh around inf

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

   10. Simplified 56.9%

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

   if 1.05e204 < 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. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. remove-double-div N/A

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

      5. metadata-eval N/A

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in ew around inf

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 66.9%

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

   9. Simplified 66.9%

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

Alternative 10: 46.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|t \cdot ew\right|\\ \mathbf{if}\;ew \leq -7.5 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.8 \cdot 10^{+210}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* t ew))))
   (if (<= ew -7.5e+188) t_1 (if (<= ew 4.8e+210) (fabs eh) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((t * ew));
	double tmp;
	if (ew <= -7.5e+188) {
		tmp = t_1;
	} else if (ew <= 4.8e+210) {
		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((t * ew))
    if (ew <= (-7.5d+188)) then
        tmp = t_1
    else if (ew <= 4.8d+210) 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((t * ew));
	double tmp;
	if (ew <= -7.5e+188) {
		tmp = t_1;
	} else if (ew <= 4.8e+210) {
		tmp = Math.abs(eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((t * ew))
	tmp = 0
	if ew <= -7.5e+188:
		tmp = t_1
	elif ew <= 4.8e+210:
		tmp = math.fabs(eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(t * ew))
	tmp = 0.0
	if (ew <= -7.5e+188)
		tmp = t_1;
	elseif (ew <= 4.8e+210)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((t * ew));
	tmp = 0.0;
	if (ew <= -7.5e+188)
		tmp = t_1;
	elseif (ew <= 4.8e+210)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -7.5e+188], t$95$1, If[LessEqual[ew, 4.8e+210], N[Abs[eh], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.4999999999999996e188 or 4.79999999999999977e210 < 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. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   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 89.5%

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

   7. Simplified 89.5%

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

   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 59.8%

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

   10. Simplified 59.8%

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

   if -7.4999999999999996e188 < ew < 4.79999999999999977e210

   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 56.5%

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

   5. Simplified 56.5%

      \[\leadsto \left|\color{blue}{eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}\right| \]
   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. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. hypot-undefine N/A

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

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

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

      13. associate-/r* N/A

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

      14. /-lowering-/.f64 N/A

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

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

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

   7. Applied egg-rr 29.0%

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

   8. Taylor expanded in eh around inf

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

   10. Simplified 56.8%

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

Alternative 11: 42.7% 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 47.0%

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

5. Simplified 47.0%

   \[\leadsto \left|\color{blue}{eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}\right| \]
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. div-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. hypot-undefine N/A

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

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

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

   13. associate-/r* N/A

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

   14. /-lowering-/.f64 N/A

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

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

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

7. Applied egg-rr 25.3%

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

8. Taylor expanded in eh around inf

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

10. Simplified 47.4%

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

Reproduce

herbie shell --seed 2024152 
(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))))))))