Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 24.4s

Alternatives: 9

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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

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. Add Preprocessing

3. Taylor expanded in t around 0 98.9%

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

4. Final simplification 98.9%

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

Alternative 3: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 61.7%

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

   3. associate-*r/ 59.7%

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

   4. hypot-1-def 67.1%

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

6. Applied egg-rr 67.1%

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

7. Taylor expanded in eh around -inf 98.6%

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

   1. associate-*r* 98.6%

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

   2. neg-mul-1 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 4: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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. associate-*l* 99.8%

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 61.7%

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

   3. associate-*r/ 59.7%

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

   4. hypot-1-def 67.1%

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

6. Applied egg-rr 67.1%

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

7. Taylor expanded in eh around inf 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 98.6% accurate, 1.5× speedup

Math

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

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(ew, Float64(sin(t) * Float64(1.0 / hypot(1.0, Float64(Float64(eh / ew) / tan(t))))), Float64(eh * cos(t))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $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. associate-*l* 99.8%

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 61.7%

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

   3. associate-*r/ 59.7%

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

   4. hypot-1-def 67.1%

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

6. Applied egg-rr 67.1%

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

7. Taylor expanded in eh around inf 98.6%

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

   1. associate-/r* 98.6%

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

   2. cos-atan 98.6%

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

   3. metadata-eval 98.6%

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

   4. hypot-undefine 98.6%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 6: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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. associate-*l* 99.8%

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 61.7%

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

   3. associate-*r/ 59.7%

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

   4. hypot-1-def 67.1%

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

6. Applied egg-rr 67.1%

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

7. Taylor expanded in eh around inf 98.6%

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

8. Taylor expanded in t around 0 98.4%

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

   1. *-commutative 80.1%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 7: 78.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 61.7%

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

   3. associate-*r/ 59.7%

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

   4. hypot-1-def 67.1%

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

6. Applied egg-rr 67.1%

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

7. Taylor expanded in t around 0 80.1%

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

   1. add-cbrt-cube 70.7%

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

   2. pow3 70.6%

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

   3. cos-atan 70.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sqrt[3]{{\left(\sin t \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}}\right)}^{3}}, eh\right)\right| \]

   4. un-div-inv 70.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sqrt[3]{{\color{blue}{\left(\frac{\sin t}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)}}^{3}}, eh\right)\right| \]

   5. hypot-1-def 70.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sqrt[3]{{\left(\frac{\sin t}{\color{blue}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}}\right)}^{3}}, eh\right)\right| \]

   6. associate-/r* 70.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sqrt[3]{{\left(\frac{\sin t}{\mathsf{hypot}\left(1, \color{blue}{\frac{\frac{eh}{ew}}{\tan t}}\right)}\right)}^{3}}, eh\right)\right| \]

9. Applied egg-rr 70.6%

   \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\sqrt[3]{{\left(\frac{\sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\right)}^{3}}}, eh\right)\right| \]
10. Step-by-step derivation

    1. rem-cbrt-cube 80.1%

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

    2. add-sqr-sqrt 50.7%

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

    3. sqrt-unprod 72.8%

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

    4. pow2 72.8%

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

    5. associate-/r* 72.8%

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

11. Applied egg-rr 72.8%

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

    1. unpow2 72.8%

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

    2. rem-sqrt-square 80.1%

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

13. Simplified 80.1%

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

14. Final simplification 80.1%

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

Alternative 8: 78.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 61.7%

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

   3. associate-*r/ 59.7%

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

   4. hypot-1-def 67.1%

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

6. Applied egg-rr 67.1%

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

7. Taylor expanded in t around 0 80.1%

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

8. Taylor expanded in t around 0 80.1%

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

   1. *-commutative 80.1%

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

10. Simplified 80.1%

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

11. Final simplification 80.1%

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

Alternative 9: 78.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(ew, \sin t, eh\right)\right| \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(fma(ew, sin(t), eh));
}

Julia

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

Wolfram

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

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

   2. fma-define 99.8%

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

   3. associate-/r* 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 61.7%

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

   3. associate-*r/ 59.7%

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

   4. hypot-1-def 67.1%

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

6. Applied egg-rr 67.1%

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

7. Taylor expanded in t around 0 80.1%

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

   1. add-cbrt-cube 70.7%

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

   2. pow3 70.6%

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

   3. cos-atan 70.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sqrt[3]{{\left(\sin t \cdot \color{blue}{\frac{1}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}}\right)}^{3}}, eh\right)\right| \]

   4. un-div-inv 70.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sqrt[3]{{\color{blue}{\left(\frac{\sin t}{\sqrt{1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right)}}^{3}}, eh\right)\right| \]

   5. hypot-1-def 70.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sqrt[3]{{\left(\frac{\sin t}{\color{blue}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}}\right)}^{3}}, eh\right)\right| \]

   6. associate-/r* 70.6%

      \[\leadsto \left|\mathsf{fma}\left(ew, \sqrt[3]{{\left(\frac{\sin t}{\mathsf{hypot}\left(1, \color{blue}{\frac{\frac{eh}{ew}}{\tan t}}\right)}\right)}^{3}}, eh\right)\right| \]

9. Applied egg-rr 70.6%

   \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\sqrt[3]{{\left(\frac{\sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\right)}^{3}}}, eh\right)\right| \]

10. Taylor expanded in eh around 0 79.9%

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

11. Final simplification 79.9%

    \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, eh\right)\right| \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(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))))))))