Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.0s

Alternatives: 9

Speedup: 1.0×

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] * ew + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t$95$1], $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 ew around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. associate-*r* N/A

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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)) + ((Math.sin(t) * ew) * Math.cos(Math.atan((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)) + ((math.sin(t) * ew) * math.cos(math.atan((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(sin(t) * ew) * cos(atan(Float64(eh / Float64(t * ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((sin(atan(((eh / ew) / tan(t)))) * (cos(t) * eh)) + ((sin(t) * ew) * cos(atan((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[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(t * ew), $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. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 3: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-atan.f64 N/A

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

   4. cos-atan N/A

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

   5. un-div-inv N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 87.5%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-atan.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-tan.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 98.2

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

7. Applied rewrites 98.2%

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

Alternative 4: 90.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|\mathsf{fma}\left(\frac{1}{\frac{1}{\sin t}}, ew, \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot t\_1\right)\right|\\ \mathbf{if}\;ew \leq -1.15 \cdot 10^{-213}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 4.8 \cdot 10^{-82}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2
         (fabs
          (fma
           (/ 1.0 (/ 1.0 (sin t)))
           ew
           (* (sin (atan (/ eh (* t ew)))) t_1)))))
   (if (<= ew -1.15e-213)
     t_2
     (if (<= ew 4.8e-82)
       (fabs (* t_1 (sin (atan (/ eh (* (tan t) ew))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs(fma((1.0 / (1.0 / sin(t))), ew, (sin(atan((eh / (t * ew)))) * t_1)));
	double tmp;
	if (ew <= -1.15e-213) {
		tmp = t_2;
	} else if (ew <= 4.8e-82) {
		tmp = fabs((t_1 * sin(atan((eh / (tan(t) * ew))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(fma(Float64(1.0 / Float64(1.0 / sin(t))), ew, Float64(sin(atan(Float64(eh / Float64(t * ew)))) * t_1)))
	tmp = 0.0
	if (ew <= -1.15e-213)
		tmp = t_2;
	elseif (ew <= 4.8e-82)
		tmp = abs(Float64(t_1 * sin(atan(Float64(eh / Float64(tan(t) * ew))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(1.0 / N[(1.0 / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew + N[(N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.15e-213], t$95$2, If[LessEqual[ew, 4.8e-82], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.15000000000000001e-213 or 4.80000000000000017e-82 < 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. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-atan.f64 N/A

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   9. Applied rewrites 97.5%

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

   10. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 90.8

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

   12. Applied rewrites 90.8%

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

   if -1.15000000000000001e-213 < ew < 4.80000000000000017e-82

   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

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.15 \cdot 10^{-213}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{1}{\frac{1}{\sin t}}, ew, \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)\right)\right|\\ \mathbf{elif}\;ew \leq 4.8 \cdot 10^{-82}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{1}{\frac{1}{\sin t}}, ew, \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\\ t_2 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+173}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -220000:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot t\_1\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left|\mathsf{fma}\left(t\_1, eh, t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ eh (* (tan t) ew))))) (t_2 (fabs (* (sin t) ew))))
   (if (<= t -2.1e+173)
     t_2
     (if (<= t -220000.0)
       (fabs (* (* (cos t) eh) t_1))
       (if (<= t 1.75e-6) (fabs (fma t_1 eh (* t ew))) t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan((eh / (tan(t) * ew))));
	double t_2 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -2.1e+173) {
		tmp = t_2;
	} else if (t <= -220000.0) {
		tmp = fabs(((cos(t) * eh) * t_1));
	} else if (t <= 1.75e-6) {
		tmp = fabs(fma(t_1, eh, (t * ew)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(eh / Float64(tan(t) * ew))))
	t_2 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -2.1e+173)
		tmp = t_2;
	elseif (t <= -220000.0)
		tmp = abs(Float64(Float64(cos(t) * eh) * t_1));
	elseif (t <= 1.75e-6)
		tmp = abs(fma(t_1, eh, Float64(t * ew)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.1e+173], t$95$2, If[LessEqual[t, -220000.0], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.75e-6], N[Abs[N[(t$95$1 * eh + N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.1e173 or 1.74999999999999997e-6 < t

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-atan.f64 N/A

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 63.0

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

   7. Applied rewrites 63.0%

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

   if -2.1e173 < t < -2.2e5

   1. Initial program 99.5%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if -2.2e5 < t < 1.74999999999999997e-6

   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. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-atan.f64 N/A

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-tan.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 96.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+173}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;t \leq -220000:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh, t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -0.0005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh, t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -0.0005)
     t_1
     (if (<= t 1.75e-6)
       (fabs (fma (sin (atan (/ eh (* (tan t) ew)))) eh (* t ew)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -0.0005) {
		tmp = t_1;
	} else if (t <= 1.75e-6) {
		tmp = fabs(fma(sin(atan((eh / (tan(t) * ew)))), eh, (t * ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -0.0005)
		tmp = t_1;
	elseif (t <= 1.75e-6)
		tmp = abs(fma(sin(atan(Float64(eh / Float64(tan(t) * ew)))), eh, Float64(t * ew)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.0005], t$95$1, If[LessEqual[t, 1.75e-6], N[Abs[N[(N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh + N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.0000000000000001e-4 or 1.74999999999999997e-6 < t

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-atan.f64 N/A

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 56.5

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

   7. Applied rewrites 56.5%

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

   if -5.0000000000000001e-4 < t < 1.74999999999999997e-6

   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. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-atan.f64 N/A

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-tan.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 97.9

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 97.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0005:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh, t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -3.6e-36) t_1 (if (<= t 1.6e-6) (fabs (- eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -3.6e-36) {
		tmp = t_1;
	} else if (t <= 1.6e-6) {
		tmp = fabs(-eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (t <= (-3.6d-36)) then
        tmp = t_1
    else if (t <= 1.6d-6) then
        tmp = abs(-eh)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (t <= -3.6e-36) {
		tmp = t_1;
	} else if (t <= 1.6e-6) {
		tmp = Math.abs(-eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if t <= -3.6e-36:
		tmp = t_1
	elif t <= 1.6e-6:
		tmp = math.fabs(-eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -3.6e-36)
		tmp = t_1;
	elseif (t <= 1.6e-6)
		tmp = abs(Float64(-eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (t <= -3.6e-36)
		tmp = t_1;
	elseif (t <= 1.6e-6)
		tmp = abs(-eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.6e-36], t$95$1, If[LessEqual[t, 1.6e-6], N[Abs[(-eh)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.60000000000000032e-36 or 1.5999999999999999e-6 < t

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-atan.f64 N/A

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 56.7

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

   7. Applied rewrites 56.7%

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

   if -3.60000000000000032e-36 < t < 1.5999999999999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-tan.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 10.3%

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

   8. Taylor expanded in eh around -inf

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

   10. Applied rewrites 79.6%

       \[\leadsto \left|-eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 45.3% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|-eh\right|\\ \mathbf{if}\;eh \leq -1.4 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.2 \cdot 10^{-228}:\\ \;\;\;\;\left|t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (- eh))))
   (if (<= eh -1.4e-131) t_1 (if (<= eh 2.2e-228) (fabs (* t ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(-eh);
	double tmp;
	if (eh <= -1.4e-131) {
		tmp = t_1;
	} else if (eh <= 2.2e-228) {
		tmp = fabs((t * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(-eh)
    if (eh <= (-1.4d-131)) then
        tmp = t_1
    else if (eh <= 2.2d-228) then
        tmp = abs((t * ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(-eh);
	double tmp;
	if (eh <= -1.4e-131) {
		tmp = t_1;
	} else if (eh <= 2.2e-228) {
		tmp = Math.abs((t * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(-eh)
	tmp = 0
	if eh <= -1.4e-131:
		tmp = t_1
	elif eh <= 2.2e-228:
		tmp = math.fabs((t * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(-eh))
	tmp = 0.0
	if (eh <= -1.4e-131)
		tmp = t_1;
	elseif (eh <= 2.2e-228)
		tmp = abs(Float64(t * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(-eh);
	tmp = 0.0;
	if (eh <= -1.4e-131)
		tmp = t_1;
	elseif (eh <= 2.2e-228)
		tmp = abs((t * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[(-eh)], $MachinePrecision]}, If[LessEqual[eh, -1.4e-131], t$95$1, If[LessEqual[eh, 2.2e-228], N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.4e-131 or 2.2e-228 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-tan.f64 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 8.9%

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

   8. Taylor expanded in eh around -inf

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

   10. Applied rewrites 50.9%

       \[\leadsto \left|-eh\right| \]

   if -1.4e-131 < eh < 2.2e-228

   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. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-atan.f64 N/A

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 85.3

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 35.8%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.4 \cdot 10^{-131}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{elif}\;eh \leq 2.2 \cdot 10^{-228}:\\ \;\;\;\;\left|t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|-eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.9% accurate, 174.0× 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(Float64(-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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-tan.f64 43.4

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

5. Applied rewrites 43.4%

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

7. Applied rewrites 7.6%

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

8. Taylor expanded in eh around -inf

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

10. Applied rewrites 43.9%

    \[\leadsto \left|-eh\right| \]
11. Add Preprocessing

Reproduce

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