Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.4s

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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

   4. *-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(ew * t), $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. Taylor expanded in t around 0 98.0%

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

3. Final simplification 98.0%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(eh * (-N[Cos[t], $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. Step-by-step derivation

   1. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.1%

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

   3. associate-*r/ 61.4%

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

   4. *-commutative 61.4%

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

   5. hypot-1-def 67.5%

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

   6. *-commutative 67.5%

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

5. Applied egg-rr 67.5%

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

   1. cos-atan 67.4%

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

   2. metadata-eval 67.4%

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

   3. associate-/l/ 67.4%

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

   4. associate-/l/ 67.4%

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

   5. hypot-udef 67.5%

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

   6. associate-/r* 67.5%

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

   7. frac-2neg 67.5%

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

   8. metadata-eval 67.5%

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

   9. frac-2neg 67.5%

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

   10. metadata-eval 67.5%

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

7. Applied egg-rr 67.5%

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

8. Taylor expanded in eh around -inf 97.5%

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

   1. associate-*r* 97.5%

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

   2. neg-mul-1 97.5%

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

10. Simplified 97.5%

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

    1. metadata-eval 97.5%

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

    2. frac-2neg 97.5%

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

    3. div-inv 97.5%

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

    4. add-sqr-sqrt 0.0%

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

    5. sqrt-unprod 97.4%

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

    6. sqr-neg 97.4%

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

    7. sqrt-unprod 97.4%

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

    8. add-sqr-sqrt 97.4%

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

    9. *-commutative 97.4%

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

12. Applied egg-rr 97.4%

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

    1. associate-*r/ 97.4%

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

    2. metadata-eval 97.4%

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

    3. associate-/l/ 97.4%

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

14. Simplified 97.4%

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

15. Final simplification 97.4%

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

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(eh * (-N[Cos[t], $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. Step-by-step derivation

   1. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.1%

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

   3. associate-*r/ 61.4%

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

   4. *-commutative 61.4%

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

   5. hypot-1-def 67.5%

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

   6. *-commutative 67.5%

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

5. Applied egg-rr 67.5%

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

   1. cos-atan 67.4%

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

   2. metadata-eval 67.4%

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

   3. associate-/l/ 67.4%

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

   4. associate-/l/ 67.4%

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

   5. hypot-udef 67.5%

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

   6. associate-/r* 67.5%

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

   7. frac-2neg 67.5%

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

   8. metadata-eval 67.5%

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

   9. frac-2neg 67.5%

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

   10. metadata-eval 67.5%

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

7. Applied egg-rr 67.5%

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

8. Taylor expanded in eh around -inf 97.5%

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

   1. associate-*r* 97.5%

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

   2. neg-mul-1 97.5%

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

10. Simplified 97.5%

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

    1. expm1-log1p-u 97.5%

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{-\left(-\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)}\right)\right)}, \left(-eh\right) \cdot \cos t\right)\right| \]

    2. expm1-udef 97.5%

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{-\left(-\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)\right)}\right)} - 1}, \left(-eh\right) \cdot \cos t\right)\right| \]

    3. remove-double-neg 97.5%

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}}\right)} - 1, \left(-eh\right) \cdot \cos t\right)\right| \]

    4. *-commutative 97.5%

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\color{blue}{\tan t \cdot ew}}\right)}\right)} - 1, \left(-eh\right) \cdot \cos t\right)\right| \]

12. Applied egg-rr 97.5%

    \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)}\right)} - 1}, \left(-eh\right) \cdot \cos t\right)\right| \]
13. Step-by-step derivation

    1. expm1-def 97.5%

       \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{\tan t \cdot ew}\right)}\right)\right)}, \left(-eh\right) \cdot \cos t\right)\right| \]

    2. expm1-log1p 97.5%

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

    3. associate-/l/ 97.5%

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

14. Simplified 97.5%

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

15. Final simplification 97.5%

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

Alternative 5: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

   4. *-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 96.8%

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

7. Final simplification 96.8%

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

Alternative 6: 98.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[(eh * (-N[Cos[t], $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. Step-by-step derivation

   1. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.1%

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

   3. associate-*r/ 61.4%

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

   4. *-commutative 61.4%

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

   5. hypot-1-def 67.5%

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

   6. *-commutative 67.5%

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

5. Applied egg-rr 67.5%

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

   1. cos-atan 67.4%

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

   2. metadata-eval 67.4%

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

   3. associate-/l/ 67.4%

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

   4. associate-/l/ 67.4%

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

   5. hypot-udef 67.5%

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

   6. associate-/r* 67.5%

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

   7. frac-2neg 67.5%

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

   8. metadata-eval 67.5%

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

   9. frac-2neg 67.5%

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

   10. metadata-eval 67.5%

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

7. Applied egg-rr 67.5%

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

8. Taylor expanded in eh around -inf 97.5%

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

   1. associate-*r* 97.5%

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

   2. neg-mul-1 97.5%

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

10. Simplified 97.5%

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

11. Taylor expanded in t around 0 96.8%

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

    1. associate-/r* 96.8%

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

13. Simplified 96.8%

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

14. Final simplification 96.8%

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

Alternative 7: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -1.25 \cdot 10^{-157}:\\ \;\;\;\;\left|\mathsf{fma}\left(t_1, \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, eh\right)\right|\\ \mathbf{elif}\;ew \leq 1.75 \cdot 10^{-41}:\\ \;\;\;\;\left|\mathsf{fma}\left(t_1, \frac{ew \cdot t}{eh}, eh \cdot \left(-\cos t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t_1, \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right), eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -1.25e-157)
     (fabs (fma t_1 (/ 1.0 (hypot 1.0 (/ eh (* ew (tan t))))) eh))
     (if (<= ew 1.75e-41)
       (fabs (fma t_1 (/ (* ew t) eh) (* eh (- (cos t)))))
       (fabs (fma t_1 (cos (atan (/ (/ eh ew) t))) eh))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (ew <= -1.25e-157) {
		tmp = fabs(fma(t_1, (1.0 / hypot(1.0, (eh / (ew * tan(t))))), eh));
	} else if (ew <= 1.75e-41) {
		tmp = fabs(fma(t_1, ((ew * t) / eh), (eh * -cos(t))));
	} else {
		tmp = fabs(fma(t_1, cos(atan(((eh / ew) / t))), eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -1.25e-157)
		tmp = abs(fma(t_1, Float64(1.0 / hypot(1.0, Float64(eh / Float64(ew * tan(t))))), eh));
	elseif (ew <= 1.75e-41)
		tmp = abs(fma(t_1, Float64(Float64(ew * t) / eh), Float64(eh * Float64(-cos(t)))));
	else
		tmp = abs(fma(t_1, cos(atan(Float64(Float64(eh / ew) / t))), eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.25e-157], N[Abs[N[(t$95$1 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + eh), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.75e-41], N[Abs[N[(t$95$1 * N[(N[(ew * t), $MachinePrecision] / eh), $MachinePrecision] + N[(eh * (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + eh), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.25000000000000005e-157

   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. Step-by-step derivation

      1. fma-def 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r* 99.8%

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

      2. sin-atan 78.8%

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

      3. associate-*r/ 77.6%

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

      4. *-commutative 77.6%

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

      5. hypot-1-def 81.1%

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

      6. *-commutative 81.1%

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

   5. Applied egg-rr 81.1%

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

   6. Taylor expanded in t around 0 82.0%

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

      1. cos-atan 81.1%

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

      2. metadata-eval 81.1%

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

      3. associate-/l/ 81.1%

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

      4. associate-/l/ 81.1%

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

      5. hypot-udef 81.1%

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

      6. associate-/r* 81.1%

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

      7. frac-2neg 81.1%

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

      8. metadata-eval 81.1%

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

      9. frac-2neg 81.1%

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

      10. metadata-eval 81.1%

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

   8. Applied egg-rr 82.0%

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

   if -1.25000000000000005e-157 < ew < 1.75e-41

   1. Initial program 99.7%

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

      1. fma-def 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r* 99.7%

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

      2. sin-atan 29.3%

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

      3. associate-*r/ 28.8%

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

      4. *-commutative 28.8%

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

      5. hypot-1-def 39.2%

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

      6. *-commutative 39.2%

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

   5. Applied egg-rr 39.2%

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

      1. cos-atan 39.2%

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

      2. metadata-eval 39.2%

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

      3. associate-/l/ 39.2%

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

      4. associate-/l/ 39.2%

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

      5. hypot-udef 39.2%

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

      6. associate-/r* 39.2%

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

      7. frac-2neg 39.2%

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

      8. metadata-eval 39.2%

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

      9. frac-2neg 39.2%

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

      10. metadata-eval 39.2%

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

   7. Applied egg-rr 39.2%

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

   8. Taylor expanded in eh around -inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. neg-mul-1 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in t around 0 87.6%

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

   if 1.75e-41 < ew

   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. Step-by-step derivation

      1. fma-def 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. sin-atan 87.7%

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

      3. associate-*r/ 87.4%

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

      4. *-commutative 87.4%

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

      5. hypot-1-def 90.7%

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

      6. *-commutative 90.7%

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

   5. Applied egg-rr 90.7%

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

   6. Taylor expanded in t around 0 84.3%

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

   7. Taylor expanded in t around 0 84.3%

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

      1. associate-/r* 84.3%

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

   9. Simplified 84.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.25 \cdot 10^{-157}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, eh\right)\right|\\ \mathbf{elif}\;ew \leq 1.75 \cdot 10^{-41}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \frac{ew \cdot t}{eh}, eh \cdot \left(-\cos t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right), eh\right)\right|\\ \end{array} \]

Alternative 8: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -3.35 \cdot 10^{-158} \lor \neg \left(ew \leq 4.8 \cdot 10^{-40}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(t_1, \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right), eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t_1, \frac{ew \cdot t}{eh}, eh \cdot \left(-\cos t\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (or (<= ew -3.35e-158) (not (<= ew 4.8e-40)))
     (fabs (fma t_1 (cos (atan (/ (/ eh ew) t))) eh))
     (fabs (fma t_1 (/ (* ew t) eh) (* eh (- (cos t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if ((ew <= -3.35e-158) || !(ew <= 4.8e-40)) {
		tmp = fabs(fma(t_1, cos(atan(((eh / ew) / t))), eh));
	} else {
		tmp = fabs(fma(t_1, ((ew * t) / eh), (eh * -cos(t))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if ((ew <= -3.35e-158) || !(ew <= 4.8e-40))
		tmp = abs(fma(t_1, cos(atan(Float64(Float64(eh / ew) / t))), eh));
	else
		tmp = abs(fma(t_1, Float64(Float64(ew * t) / eh), Float64(eh * Float64(-cos(t)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -3.35e-158], N[Not[LessEqual[ew, 4.8e-40]], $MachinePrecision]], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[(N[(ew * t), $MachinePrecision] / eh), $MachinePrecision] + N[(eh * (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.35e-158 or 4.79999999999999982e-40 < 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. Step-by-step derivation

      1. fma-def 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r* 99.8%

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

      2. sin-atan 82.4%

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

      3. associate-*r/ 81.6%

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

      4. *-commutative 81.6%

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

      5. hypot-1-def 85.0%

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

      6. *-commutative 85.0%

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

   5. Applied egg-rr 85.0%

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

   6. Taylor expanded in t around 0 82.9%

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

   7. Taylor expanded in t around 0 82.9%

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

      1. associate-/r* 82.9%

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

   9. Simplified 82.9%

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

   if -3.35e-158 < ew < 4.79999999999999982e-40

   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. Step-by-step derivation

      1. fma-def 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r* 99.7%

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

      2. sin-atan 29.3%

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

      3. associate-*r/ 28.8%

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

      4. *-commutative 28.8%

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

      5. hypot-1-def 39.2%

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

      6. *-commutative 39.2%

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

   5. Applied egg-rr 39.2%

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

      1. cos-atan 39.2%

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

      2. metadata-eval 39.2%

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

      3. associate-/l/ 39.2%

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

      4. associate-/l/ 39.2%

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

      5. hypot-udef 39.2%

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

      6. associate-/r* 39.2%

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

      7. frac-2neg 39.2%

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

      8. metadata-eval 39.2%

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

      9. frac-2neg 39.2%

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

      10. metadata-eval 39.2%

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

   7. Applied egg-rr 39.2%

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

   8. Taylor expanded in eh around -inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. neg-mul-1 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in t around 0 87.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.35 \cdot 10^{-158} \lor \neg \left(ew \leq 4.8 \cdot 10^{-40}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right), eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \frac{ew \cdot t}{eh}, eh \cdot \left(-\cos t\right)\right)\right|\\ \end{array} \]

Alternative 9: 69.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-6} \lor \neg \left(t \leq 0.001\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \frac{ew \cdot t}{eh}, eh \cdot \left(-\cos t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh + \frac{ew \cdot t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -6e-6) (not (<= t 0.001)))
   (fabs (fma (* ew (sin t)) (/ (* ew t) eh) (* eh (- (cos t)))))
   (fabs (+ eh (/ (* ew t) (hypot 1.0 (/ eh (* ew (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -6e-6) || !(t <= 0.001)) {
		tmp = fabs(fma((ew * sin(t)), ((ew * t) / eh), (eh * -cos(t))));
	} else {
		tmp = fabs((eh + ((ew * t) / hypot(1.0, (eh / (ew * tan(t)))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -6e-6) || !(t <= 0.001))
		tmp = abs(fma(Float64(ew * sin(t)), Float64(Float64(ew * t) / eh), Float64(eh * Float64(-cos(t)))));
	else
		tmp = abs(Float64(eh + Float64(Float64(ew * t) / hypot(1.0, Float64(eh / Float64(ew * tan(t)))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -6e-6], N[Not[LessEqual[t, 0.001]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(N[(ew * t), $MachinePrecision] / eh), $MachinePrecision] + N[(eh * (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh + N[(N[(ew * t), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.0000000000000002e-6 or 1e-3 < 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. Step-by-step derivation

      1. fma-def 99.6%

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

      2. associate-/l/ 99.6%

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

      3. associate-*l* 99.6%

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

      4. associate-/l/ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r* 99.6%

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

      2. sin-atan 72.5%

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

      3. associate-*r/ 71.4%

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

      4. *-commutative 71.4%

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

      5. hypot-1-def 77.1%

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

      6. *-commutative 77.1%

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

   5. Applied egg-rr 77.1%

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

      1. cos-atan 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-/l/ 77.1%

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

      4. associate-/l/ 77.1%

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

      5. hypot-udef 77.1%

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

      6. associate-/r* 77.1%

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

      7. frac-2neg 77.1%

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

      8. metadata-eval 77.1%

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

      9. frac-2neg 77.1%

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

      10. metadata-eval 77.1%

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

   7. Applied egg-rr 77.1%

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

   8. Taylor expanded in eh around -inf 96.8%

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

      1. associate-*r* 96.8%

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

      2. neg-mul-1 96.8%

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

   10. Simplified 96.8%

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

   11. Taylor expanded in t around 0 42.6%

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

   if -6.0000000000000002e-6 < t < 1e-3

   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. Step-by-step derivation

      1. fma-def 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r* 100.0%

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

      2. sin-atan 50.8%

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

      3. associate-*r/ 50.6%

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

      4. *-commutative 50.6%

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

      5. hypot-1-def 57.0%

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

      6. *-commutative 57.0%

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

   5. Applied egg-rr 57.0%

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

   6. Taylor expanded in t around 0 98.1%

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

   7. Taylor expanded in t around 0 98.1%

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

      1. fma-udef 98.1%

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

      2. cos-atan 98.1%

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

      3. hypot-1-def 98.1%

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

      4. *-commutative 98.1%

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

      5. un-div-inv 98.1%

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

   9. Applied egg-rr 98.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-6} \lor \neg \left(t \leq 0.001\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \sin t, \frac{ew \cdot t}{eh}, eh \cdot \left(-\cos t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh + \frac{ew \cdot t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \end{array} \]

Alternative 10: 56.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh + N[(N[(ew * t), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 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. Step-by-step derivation

   1. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.1%

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

   3. associate-*r/ 61.4%

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

   4. *-commutative 61.4%

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

   5. hypot-1-def 67.5%

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

   6. *-commutative 67.5%

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

5. Applied egg-rr 67.5%

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

6. Taylor expanded in t around 0 75.2%

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

7. Taylor expanded in t around 0 53.0%

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

   1. fma-udef 53.0%

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

   2. cos-atan 53.5%

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

   3. hypot-1-def 53.5%

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

   4. *-commutative 53.5%

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

   5. un-div-inv 53.5%

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

9. Applied egg-rr 53.5%

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

10. Final simplification 53.5%

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

Alternative 11: 42.9% 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. Step-by-step derivation

   1. fma-def 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 62.1%

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

   3. associate-*r/ 61.4%

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

   4. *-commutative 61.4%

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

   5. hypot-1-def 67.5%

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

   6. *-commutative 67.5%

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

5. Applied egg-rr 67.5%

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

6. Taylor expanded in t around 0 75.2%

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

7. Taylor expanded in t around 0 53.0%

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

8. Taylor expanded in ew around 0 42.2%

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

9. Final simplification 42.2%

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

Reproduce

herbie shell --seed 2023338 
(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))))))))