Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 27.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 49.9%

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

   2. sqrt-unprod 96.8%

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

   3. sqr-neg 96.8%

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

   4. sqrt-unprod 49.9%

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

   5. add-sqr-sqrt 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. expm1-log1p-u 80.3%

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

   2. expm1-udef 79.8%

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

   3. add-sqr-sqrt 39.3%

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

   4. sqrt-unprod 74.3%

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

   5. sqr-neg 74.3%

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

   6. sqrt-unprod 40.5%

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

   7. add-sqr-sqrt 83.3%

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

3. Applied egg-rr 83.3%

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

   1. expm1-def 83.8%

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

   2. expm1-log1p 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cos-atan 99.8%

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

   2. hypot-1-def 99.8%

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

   3. associate-/l* 99.8%

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

   4. add-sqr-sqrt 49.9%

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

   5. sqrt-unprod 94.2%

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

   6. sqr-neg 94.2%

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

   7. sqrt-unprod 49.9%

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

   8. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 99.2%

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

   1. mul-1-neg 99.2%

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

   2. distribute-rgt-neg-in 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 5: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 49.9%

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

   2. sqrt-unprod 96.8%

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

   3. sqr-neg 96.8%

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

   4. sqrt-unprod 49.9%

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

   5. add-sqr-sqrt 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 79.1%

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

   3. associate-*r/ 77.5%

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

   4. associate-*r/ 77.6%

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

   5. distribute-lft-neg-out 77.6%

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

   6. distribute-rgt-neg-out 77.6%

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

   7. *-commutative 77.6%

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

   8. associate-/l* 77.6%

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

   9. add-sqr-sqrt 40.3%

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

   10. sqrt-unprod 67.3%

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

   11. sqr-neg 67.3%

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

   12. sqrt-unprod 36.9%

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

   13. add-sqr-sqrt 76.6%

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

   14. hypot-1-def 80.7%

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

7. Applied egg-rr 84.1%

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

   1. associate-/l* 92.1%

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

   2. associate-/r/ 92.1%

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

   3. associate-/r/ 87.9%

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

   4. *-commutative 87.9%

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

   5. associate-/r/ 87.8%

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

   6. *-commutative 87.8%

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

9. Simplified 87.8%

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

10. Taylor expanded in eh around -inf 98.4%

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

    1. associate-*r* 98.4%

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

    2. neg-mul-1 98.4%

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

    3. *-commutative 98.4%

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

12. Simplified 98.4%

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

13. Final simplification 98.4%

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

Alternative 6: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 49.9%

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

   2. sqrt-unprod 96.8%

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

   3. sqr-neg 96.8%

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

   4. sqrt-unprod 49.9%

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

   5. add-sqr-sqrt 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-/l* 99.8%

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

   2. cos-atan 99.8%

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

   3. hypot-1-def 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in eh around 0 98.6%

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

9. Final simplification 98.6%

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

Alternative 7: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 49.9%

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

   2. sqrt-unprod 96.8%

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

   3. sqr-neg 96.8%

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

   4. sqrt-unprod 49.9%

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

   5. add-sqr-sqrt 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r* 99.8%

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

   2. sin-atan 79.1%

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

   3. associate-*r/ 77.5%

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

   4. associate-*r/ 77.6%

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

   5. distribute-lft-neg-out 77.6%

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

   6. distribute-rgt-neg-out 77.6%

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

   7. *-commutative 77.6%

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

   8. associate-/l* 77.6%

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

   9. add-sqr-sqrt 40.3%

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

   10. sqrt-unprod 67.3%

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

   11. sqr-neg 67.3%

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

   12. sqrt-unprod 36.9%

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

   13. add-sqr-sqrt 76.6%

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

   14. hypot-1-def 80.7%

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

7. Applied egg-rr 84.1%

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

   1. associate-/l* 92.1%

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

   2. associate-/r/ 92.1%

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

   3. associate-/r/ 87.9%

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

   4. *-commutative 87.9%

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

   5. associate-/r/ 87.8%

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

   6. *-commutative 87.8%

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

9. Simplified 87.8%

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

10. Taylor expanded in eh around inf 98.4%

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

11. Final simplification 98.4%

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

Alternative 8: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.9 \cdot 10^{-156} \lor \neg \left(eh \leq 10^{-75}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \left(-\cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right), eh \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.9e-156) (not (<= eh 1e-75)))
   (fabs (fma ew (* (- (cos t)) (cos (atan (/ (* eh t) ew)))) (* eh (sin t))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.9e-156) || !(eh <= 1e-75)) {
		tmp = fabs(fma(ew, (-cos(t) * cos(atan(((eh * t) / ew)))), (eh * sin(t))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.9e-156) || !(eh <= 1e-75))
		tmp = abs(fma(ew, Float64(Float64(-cos(t)) * cos(atan(Float64(Float64(eh * t) / ew)))), Float64(eh * sin(t))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.9e-156], N[Not[LessEqual[eh, 1e-75]], $MachinePrecision]], N[Abs[N[(ew * N[((-N[Cos[t], $MachinePrecision]) * N[Cos[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.9000000000000001e-156 or 9.9999999999999996e-76 < eh

   1. Initial program 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-def 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 50.7%

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

      2. sqrt-unprod 95.3%

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

      3. sqr-neg 95.3%

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

      4. sqrt-unprod 49.0%

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

      5. add-sqr-sqrt 99.7%

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

      6. associate-*r/ 99.7%

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

      7. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r* 99.7%

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

      2. sin-atan 69.5%

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

      3. associate-*r/ 67.1%

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

      4. associate-*r/ 67.3%

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

      5. distribute-lft-neg-out 67.3%

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

      6. distribute-rgt-neg-out 67.3%

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

      7. *-commutative 67.3%

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

      8. associate-/l* 67.3%

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

      9. add-sqr-sqrt 40.1%

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

      10. sqrt-unprod 53.3%

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

      11. sqr-neg 53.3%

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

      12. sqrt-unprod 26.6%

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

      13. add-sqr-sqrt 65.8%

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

      14. hypot-1-def 71.8%

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

   7. Applied egg-rr 76.8%

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

      1. associate-/l* 88.5%

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

      2. associate-/r/ 88.5%

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

      3. associate-/r/ 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-/r/ 82.2%

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

      6. *-commutative 82.2%

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

   9. Simplified 82.2%

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

   10. Taylor expanded in eh around inf 97.8%

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

   11. Taylor expanded in t around 0 89.8%

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

   if -3.9000000000000001e-156 < eh < 9.9999999999999996e-76

   1. Initial program 99.9%

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

      1. fabs-sub 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 96.5%

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

      1. +-inverses 96.5%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \frac{eh \cdot \color{blue}{0}}{2}\right)\right| \]

      2. *-commutative 96.5%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \frac{\color{blue}{0 \cdot eh}}{2}\right)\right| \]

      3. associate-/l* 96.5%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{\frac{0}{\frac{2}{eh}}}\right)\right| \]

      4. div0 96.5%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{0}\right)\right| \]

   7. Simplified 96.5%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{0}\right)\right| \]
   8. Step-by-step derivation

      1. log1p-expm1-u 96.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right)\right)\right)}, 0\right)\right| \]

   9. Applied egg-rr 96.4%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right)\right)\right)}, 0\right)\right| \]

   11. Applied egg-rr 57.0%

       \[\leadsto \left|\color{blue}{\frac{{\left(ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}^{2} - 0}{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - 0}}\right| \]
   12. Step-by-step derivation

       1. --rgt-identity 57.0%

          \[\leadsto \left|\frac{\color{blue}{{\left(ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}^{2}}}{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - 0}\right| \]

       2. --rgt-identity 57.0%

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

   13. Simplified 57.0%

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

   14. Taylor expanded in ew around inf 96.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.9 \cdot 10^{-156} \lor \neg \left(eh \leq 10^{-75}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \left(-\cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right), eh \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]

Alternative 9: 62.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \cos t\right| \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(ew * cos(t)))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

5. Applied egg-rr 65.7%

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

   1. +-inverses 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \frac{eh \cdot \color{blue}{0}}{2}\right)\right| \]

   2. *-commutative 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \frac{\color{blue}{0 \cdot eh}}{2}\right)\right| \]

   3. associate-/l* 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{\frac{0}{\frac{2}{eh}}}\right)\right| \]

   4. div0 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{0}\right)\right| \]

7. Simplified 65.7%

   \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{0}\right)\right| \]
8. Step-by-step derivation

   1. log1p-expm1-u 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right)\right)\right)}, 0\right)\right| \]

9. Applied egg-rr 65.7%

   \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right)\right)\right)}, 0\right)\right| \]

11. Applied egg-rr 39.6%

    \[\leadsto \left|\color{blue}{\frac{{\left(ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}^{2} - 0}{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - 0}}\right| \]
12. Step-by-step derivation

    1. --rgt-identity 39.6%

       \[\leadsto \left|\frac{\color{blue}{{\left(ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}^{2}}}{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - 0}\right| \]

    2. --rgt-identity 39.6%

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

13. Simplified 39.6%

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

14. Taylor expanded in ew around inf 65.8%

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

15. Final simplification 65.8%

    \[\leadsto \left|ew \cdot \cos t\right| \]

Alternative 10: 43.2% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(ew)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

5. Applied egg-rr 65.7%

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

   1. +-inverses 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \frac{eh \cdot \color{blue}{0}}{2}\right)\right| \]

   2. *-commutative 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \frac{\color{blue}{0 \cdot eh}}{2}\right)\right| \]

   3. associate-/l* 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{\frac{0}{\frac{2}{eh}}}\right)\right| \]

   4. div0 65.7%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{0}\right)\right| \]

7. Simplified 65.7%

   \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right) \cdot \left(-\cos t\right), \color{blue}{0}\right)\right| \]

8. Taylor expanded in t around 0 44.4%

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

   1. mul-1-neg 44.4%

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

   2. *-commutative 44.4%

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

   3. mul-1-neg 44.4%

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

   4. associate-*l/ 44.4%

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

   5. *-commutative 44.4%

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

   6. distribute-lft-neg-out 44.4%

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

   7. distribute-rgt-neg-in 44.4%

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

   8. distribute-lft-neg-out 44.4%

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

   9. distribute-rgt-neg-in 44.4%

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

10. Simplified 44.4%

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

    1. expm1-log1p-u 28.1%

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

    2. expm1-udef 15.1%

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

12. Applied egg-rr 16.1%

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

    1. expm1-def 29.0%

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

    2. expm1-log1p 44.1%

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

    3. associate-*r/ 44.1%

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

    4. *-commutative 44.1%

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

    5. *-lft-identity 44.1%

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

    6. times-frac 44.1%

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

    7. /-rgt-identity 44.1%

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

14. Simplified 44.1%

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

15. Taylor expanded in ew around inf 44.5%

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

16. Final simplification 44.5%

    \[\leadsto \left|ew\right| \]

Reproduce

herbie shell --seed 2023277 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))