Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.8s

Alternatives: 10

Speedup: N/A×

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, 1.1× speedup

Math

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

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(Float64(eh / 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[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $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. frac-2neg 99.8%

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

   2. distribute-lft-neg-out 99.8%

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

   3. remove-double-neg 99.8%

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

   4. associate-*l/ 99.8%

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

   5. *-commutative 99.8%

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

   6. cos-atan 99.8%

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

   7. frac-2neg 99.8%

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

   8. metadata-eval 99.8%

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

   9. div-inv 99.8%

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

   10. metadata-eval 99.8%

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

   11. associate-*l/ 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. metadata-eval 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-*l/ 99.8%

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

   5. *-commutative 99.8%

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

5. Simplified 99.8%

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

   1. div-inv 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/r/ 99.8%

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

   4. hypot-udef 99.8%

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

   5. metadata-eval 99.8%

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

   6. distribute-frac-neg 99.8%

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

   7. distribute-frac-neg 99.8%

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

   8. sqr-neg 99.8%

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

   9. hypot-1-def 99.8%

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

   10. associate-/r/ 99.8%

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

   11. *-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. *-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 98.9% 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 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-*r* 98.6%

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

   3. neg-mul-1 98.6%

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

4. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 98.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. cancel-sign-sub-inv 99.8%

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

   4. associate-*l* 99.8%

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

   5. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 78.0%

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

   3. associate-/l* 77.9%

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

   4. associate-*l/ 75.0%

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

   5. div-inv 75.0%

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

   6. clear-num 75.0%

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

5. Applied egg-rr 85.4%

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

6. Taylor expanded in eh around -inf 97.4%

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

7. Final simplification 97.4%

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

Alternative 4: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\\ \mathbf{if}\;ew \leq -2.2 \cdot 10^{+22}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t, \frac{-ew}{\frac{\mathsf{hypot}\left(1, eh \cdot \frac{-\tan t}{ew}\right)}{\cos t}}\right)\right|\\ \mathbf{elif}\;ew \leq 2.4 \cdot 10^{+82}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \sin t \cdot \sin t_1, -ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t, ew \cdot \left(\cos t \cdot \left(-\cos t_1\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (tan t) (/ eh (- ew))))))
   (if (<= ew -2.2e+22)
     (fabs
      (fma eh t (/ (- ew) (/ (hypot 1.0 (* eh (/ (- (tan t)) ew))) (cos t)))))
     (if (<= ew 2.4e+82)
       (fabs (fma eh (* (sin t) (sin t_1)) (- ew)))
       (fabs (fma eh t (* ew (* (cos t) (- (cos t_1))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((tan(t) * (eh / -ew)));
	double tmp;
	if (ew <= -2.2e+22) {
		tmp = fabs(fma(eh, t, (-ew / (hypot(1.0, (eh * (-tan(t) / ew))) / cos(t)))));
	} else if (ew <= 2.4e+82) {
		tmp = fabs(fma(eh, (sin(t) * sin(t_1)), -ew));
	} else {
		tmp = fabs(fma(eh, t, (ew * (cos(t) * -cos(t_1)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(tan(t) * Float64(eh / Float64(-ew))))
	tmp = 0.0
	if (ew <= -2.2e+22)
		tmp = abs(fma(eh, t, Float64(Float64(-ew) / Float64(hypot(1.0, Float64(eh * Float64(Float64(-tan(t)) / ew))) / cos(t)))));
	elseif (ew <= 2.4e+82)
		tmp = abs(fma(eh, Float64(sin(t) * sin(t_1)), Float64(-ew)));
	else
		tmp = abs(fma(eh, t, Float64(ew * Float64(cos(t) * Float64(-cos(t_1))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.2e+22], N[Abs[N[(eh * t + N[((-ew) / N[(N[Sqrt[1.0 ^ 2 + N[(eh * N[((-N[Tan[t], $MachinePrecision]) / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.4e+82], N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] + (-ew)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * t + N[(ew * N[(N[Cos[t], $MachinePrecision] * (-N[Cos[t$95$1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.2e22

   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. associate-*l* 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. sin-atan 98.3%

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

      3. associate-/l* 98.3%

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

      4. associate-*l/ 98.3%

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

      5. div-inv 98.2%

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

      6. clear-num 98.2%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in eh around -inf 97.9%

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

      1. associate-*r* 97.9%

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

      2. cos-atan 97.9%

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

      3. un-div-inv 97.9%

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

      4. distribute-rgt-neg-out 97.9%

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

      5. distribute-lft-neg-in 97.9%

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

      6. associate-/l* 97.9%

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

   8. Applied egg-rr 97.9%

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

   9. Taylor expanded in t around 0 80.4%

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

   if -2.2e22 < ew < 2.39999999999999998e82

   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. associate-*l* 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. neg-sub0 99.8%

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

      4. flip-- 99.8%

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

      5. +-lft-identity 99.8%

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

      6. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 87.8%

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

      1. mul-1-neg 87.8%

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

   10. Simplified 87.8%

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

   if 2.39999999999999998e82 < ew

   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. associate-*l* 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. sin-atan 99.8%

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

      3. associate-/l* 99.8%

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

      4. associate-*l/ 99.8%

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

      5. div-inv 99.8%

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

      6. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in eh around -inf 99.8%

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

   7. Taylor expanded in t around 0 88.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.2 \cdot 10^{+22}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t, \frac{-ew}{\frac{\mathsf{hypot}\left(1, eh \cdot \frac{-\tan t}{ew}\right)}{\cos t}}\right)\right|\\ \mathbf{elif}\;ew \leq 2.4 \cdot 10^{+82}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right), -ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t, ew \cdot \left(\cos t \cdot \left(-\cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)\right)\right)\right)\right|\\ \end{array} \]

Alternative 5: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.35 \cdot 10^{+22} \lor \neg \left(ew \leq 4.4 \cdot 10^{+84}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t, \frac{-ew}{\frac{\mathsf{hypot}\left(1, eh \cdot \frac{-\tan t}{ew}\right)}{\cos t}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right), -ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.35e+22) (not (<= ew 4.4e+84)))
   (fabs
    (fma eh t (/ (- ew) (/ (hypot 1.0 (* eh (/ (- (tan t)) ew))) (cos t)))))
   (fabs (fma eh (* (sin t) (sin (atan (* (tan t) (/ eh (- ew)))))) (- ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.35e+22) || !(ew <= 4.4e+84)) {
		tmp = fabs(fma(eh, t, (-ew / (hypot(1.0, (eh * (-tan(t) / ew))) / cos(t)))));
	} else {
		tmp = fabs(fma(eh, (sin(t) * sin(atan((tan(t) * (eh / -ew))))), -ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2.35e+22) || !(ew <= 4.4e+84))
		tmp = abs(fma(eh, t, Float64(Float64(-ew) / Float64(hypot(1.0, Float64(eh * Float64(Float64(-tan(t)) / ew))) / cos(t)))));
	else
		tmp = abs(fma(eh, Float64(sin(t) * sin(atan(Float64(tan(t) * Float64(eh / Float64(-ew)))))), Float64(-ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.35e+22], N[Not[LessEqual[ew, 4.4e+84]], $MachinePrecision]], N[Abs[N[(eh * t + N[((-ew) / N[(N[Sqrt[1.0 ^ 2 + N[(eh * N[((-N[Tan[t], $MachinePrecision]) / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-ew)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.3500000000000001e22 or 4.3999999999999997e84 < ew

   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. associate-*l* 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. sin-atan 98.9%

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

      3. associate-/l* 98.9%

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

      4. associate-*l/ 98.9%

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

      5. div-inv 98.9%

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

      6. clear-num 98.9%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in eh around -inf 98.7%

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

      1. associate-*r* 98.7%

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

      2. cos-atan 98.7%

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

      3. un-div-inv 98.7%

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

      4. distribute-rgt-neg-out 98.7%

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

      5. distribute-lft-neg-in 98.7%

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

      6. associate-/l* 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in t around 0 83.7%

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

   if -2.3500000000000001e22 < ew < 4.3999999999999997e84

   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. associate-*l* 99.8%

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

      3. cancel-sign-sub-inv 99.8%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. neg-sub0 99.8%

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

      4. flip-- 99.8%

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

      5. +-lft-identity 99.8%

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

      6. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 87.8%

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

      1. mul-1-neg 87.8%

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

   10. Simplified 87.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.35 \cdot 10^{+22} \lor \neg \left(ew \leq 4.4 \cdot 10^{+84}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t, \frac{-ew}{\frac{\mathsf{hypot}\left(1, eh \cdot \frac{-\tan t}{ew}\right)}{\cos t}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right), -ew\right)\right|\\ \end{array} \]

Alternative 6: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. cancel-sign-sub-inv 99.8%

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

   4. associate-*l* 99.8%

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

   5. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. sin-atan 78.0%

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

   3. associate-/l* 77.9%

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

   4. associate-*l/ 75.0%

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

   5. div-inv 75.0%

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

   6. clear-num 75.0%

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

5. Applied egg-rr 85.4%

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

6. Taylor expanded in eh around -inf 93.5%

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

   1. associate-*r* 93.5%

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

   2. cos-atan 93.4%

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

   3. un-div-inv 93.4%

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

   4. distribute-rgt-neg-out 93.4%

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

   5. distribute-lft-neg-in 93.4%

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

   6. associate-/l* 93.4%

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

8. Applied egg-rr 93.4%

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

9. Taylor expanded in t around inf 97.4%

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

10. Final simplification 97.4%

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

Alternative 7: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\tan t\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+220} \lor \neg \left(t \leq -2.5 \cdot 10^{+75}\right) \land \left(t \leq -1 \lor \neg \left(t \leq 1.6 \cdot 10^{+160}\right)\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot t_1}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t, \frac{-ew}{\frac{\mathsf{hypot}\left(1, eh \cdot \frac{t_1}{ew}\right)}{\cos t}}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (tan t))))
   (if (or (<= t -4.5e+220)
           (and (not (<= t -2.5e+75)) (or (<= t -1.0) (not (<= t 1.6e+160)))))
     (fabs (* (* eh (sin t)) (sin (atan (/ (* eh t_1) ew)))))
     (fabs (fma eh t (/ (- ew) (/ (hypot 1.0 (* eh (/ t_1 ew))) (cos t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -tan(t);
	double tmp;
	if ((t <= -4.5e+220) || (!(t <= -2.5e+75) && ((t <= -1.0) || !(t <= 1.6e+160)))) {
		tmp = fabs(((eh * sin(t)) * sin(atan(((eh * t_1) / ew)))));
	} else {
		tmp = fabs(fma(eh, t, (-ew / (hypot(1.0, (eh * (t_1 / ew))) / cos(t)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(-tan(t))
	tmp = 0.0
	if ((t <= -4.5e+220) || (!(t <= -2.5e+75) && ((t <= -1.0) || !(t <= 1.6e+160))))
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * t_1) / ew)))));
	else
		tmp = abs(fma(eh, t, Float64(Float64(-ew) / Float64(hypot(1.0, Float64(eh * Float64(t_1 / ew))) / cos(t)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Tan[t], $MachinePrecision])}, If[Or[LessEqual[t, -4.5e+220], And[N[Not[LessEqual[t, -2.5e+75]], $MachinePrecision], Or[LessEqual[t, -1.0], N[Not[LessEqual[t, 1.6e+160]], $MachinePrecision]]]], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * t$95$1), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * t + N[((-ew) / N[(N[Sqrt[1.0 ^ 2 + N[(eh * N[(t$95$1 / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.50000000000000011e220 or -2.5000000000000001e75 < t < -1 or 1.5999999999999999e160 < t

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. associate-*l* 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

      4. associate-*l* 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. neg-sub0 99.6%

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

      4. flip-- 99.6%

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

      5. +-lft-identity 99.6%

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

      6. associate-*r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.7%

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

      3. associate-*r/ 99.7%

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

      4. associate-/l* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 72.4%

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

      1. mul-1-neg 72.4%

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

   10. Simplified 72.4%

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

   11. Taylor expanded in eh around inf 68.0%

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

       1. associate-*r* 68.0%

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

       2. associate-*r/ 68.0%

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

       3. mul-1-neg 68.0%

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

       4. distribute-lft-neg-out 68.0%

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

       5. *-commutative 68.0%

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

   13. Simplified 68.0%

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

   if -4.50000000000000011e220 < t < -2.5000000000000001e75 or -1 < t < 1.5999999999999999e160

   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. associate-*l* 99.9%

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

      3. cancel-sign-sub-inv 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. sin-atan 84.0%

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

      3. associate-/l* 84.0%

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

      4. associate-*l/ 80.0%

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

      5. div-inv 80.0%

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

      6. clear-num 80.0%

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

   5. Applied egg-rr 87.7%

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

   6. Taylor expanded in eh around -inf 92.5%

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

      1. associate-*r* 92.5%

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

      2. cos-atan 92.4%

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

      3. un-div-inv 92.4%

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

      4. distribute-rgt-neg-out 92.4%

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

      5. distribute-lft-neg-in 92.4%

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

      6. associate-/l* 92.4%

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

   8. Applied egg-rr 92.4%

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

   9. Taylor expanded in t around 0 84.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+220} \lor \neg \left(t \leq -2.5 \cdot 10^{+75}\right) \land \left(t \leq -1 \lor \neg \left(t \leq 1.6 \cdot 10^{+160}\right)\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, t, \frac{-ew}{\frac{\mathsf{hypot}\left(1, eh \cdot \frac{-\tan t}{ew}\right)}{\cos t}}\right)\right|\\ \end{array} \]

Alternative 8: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\\ \mathbf{if}\;t \leq -0.0037 \lor \neg \left(t \leq 0.02\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \left(t \cdot eh\right) - ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ (* eh (- (tan t))) ew)))))
   (if (or (<= t -0.0037) (not (<= t 0.02)))
     (fabs (* (* eh (sin t)) t_1))
     (fabs (- (* t_1 (* t eh)) ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan(((eh * -tan(t)) / ew)));
	double tmp;
	if ((t <= -0.0037) || !(t <= 0.02)) {
		tmp = fabs(((eh * sin(t)) * t_1));
	} else {
		tmp = fabs(((t_1 * (t * eh)) - ew));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(atan(((eh * -tan(t)) / ew)))
    if ((t <= (-0.0037d0)) .or. (.not. (t <= 0.02d0))) then
        tmp = abs(((eh * sin(t)) * t_1))
    else
        tmp = abs(((t_1 * (t * eh)) - ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan(((eh * -Math.tan(t)) / ew)));
	double tmp;
	if ((t <= -0.0037) || !(t <= 0.02)) {
		tmp = Math.abs(((eh * Math.sin(t)) * t_1));
	} else {
		tmp = Math.abs(((t_1 * (t * eh)) - ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan(((eh * -math.tan(t)) / ew)))
	tmp = 0
	if (t <= -0.0037) or not (t <= 0.02):
		tmp = math.fabs(((eh * math.sin(t)) * t_1))
	else:
		tmp = math.fabs(((t_1 * (t * eh)) - ew))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(Float64(eh * Float64(-tan(t))) / ew)))
	tmp = 0.0
	if ((t <= -0.0037) || !(t <= 0.02))
		tmp = abs(Float64(Float64(eh * sin(t)) * t_1));
	else
		tmp = abs(Float64(Float64(t_1 * Float64(t * eh)) - ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan(((eh * -tan(t)) / ew)));
	tmp = 0.0;
	if ((t <= -0.0037) || ~((t <= 0.02)))
		tmp = abs(((eh * sin(t)) * t_1));
	else
		tmp = abs(((t_1 * (t * eh)) - ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(N[(eh * (-N[Tan[t], $MachinePrecision])), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t, -0.0037], N[Not[LessEqual[t, 0.02]], $MachinePrecision]], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$1 * N[(t * eh), $MachinePrecision]), $MachinePrecision] - ew), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0037000000000000002 or 0.0200000000000000004 < t

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. associate-*l* 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

      4. associate-*l* 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. neg-sub0 99.6%

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

      4. flip-- 99.6%

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

      5. +-lft-identity 99.6%

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

      6. associate-*r/ 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/l* 99.6%

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

      3. associate-*r/ 99.6%

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

      4. associate-/l* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in t around 0 59.2%

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

      1. mul-1-neg 59.2%

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

   10. Simplified 59.2%

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

   11. Taylor expanded in eh around inf 51.4%

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

       1. associate-*r* 51.4%

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

       2. associate-*r/ 51.4%

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

       3. mul-1-neg 51.4%

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

       4. distribute-lft-neg-out 51.4%

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

       5. *-commutative 51.4%

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

   13. Simplified 51.4%

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

   if -0.0037000000000000002 < t < 0.0200000000000000004

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. associate-*l* 100.0%

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

      3. cancel-sign-sub-inv 100.0%

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

      4. associate-*l* 100.0%

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

      5. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. neg-sub0 100.0%

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

      4. flip-- 100.0%

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

      5. +-lft-identity 100.0%

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

      6. associate-*r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 97.4%

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

      1. mul-1-neg 97.4%

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

   10. Simplified 97.4%

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

   11. Taylor expanded in t around 0 97.4%

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

       1. mul-1-neg 97.4%

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

       2. +-commutative 97.4%

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

       3. unsub-neg 97.4%

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

       4. associate-*r* 97.4%

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

       5. associate-*r/ 97.4%

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

       6. mul-1-neg 97.4%

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

       7. distribute-lft-neg-out 97.4%

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

       8. *-commutative 97.4%

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

   13. Simplified 97.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0037 \lor \neg \left(t \leq 0.02\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh \cdot \left(-\tan t\right)}{ew}\right) \cdot \left(t \cdot eh\right) - ew\right|\\ \end{array} \]

Alternative 9: 55.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. cancel-sign-sub-inv 99.8%

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

   4. associate-*l* 99.8%

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

   5. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r* 99.8%

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

   3. neg-sub0 99.8%

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

   4. flip-- 99.8%

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

   5. +-lft-identity 99.8%

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

   6. associate-*r/ 99.7%

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

5. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/l* 99.7%

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

   3. associate-*r/ 99.7%

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

   4. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 78.2%

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

   1. mul-1-neg 78.2%

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

10. Simplified 78.2%

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

11. Taylor expanded in t around 0 54.6%

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

    1. mul-1-neg 54.6%

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

    2. +-commutative 54.6%

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

    3. unsub-neg 54.6%

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

    4. associate-*r* 54.5%

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

    5. associate-*r/ 54.5%

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

    6. mul-1-neg 54.5%

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

    7. distribute-lft-neg-out 54.5%

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

    8. *-commutative 54.5%

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

13. Simplified 54.5%

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

14. Final simplification 54.5%

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

Alternative 10: 42.7% 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. associate-*l* 99.8%

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

   3. cancel-sign-sub-inv 99.8%

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

   4. associate-*l* 99.8%

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

   5. fma-def 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r* 99.8%

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

   3. neg-sub0 99.8%

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

   4. flip-- 99.8%

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

   5. +-lft-identity 99.8%

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

   6. associate-*r/ 99.7%

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

5. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/l* 99.7%

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

   3. associate-*r/ 99.7%

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

   4. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 78.2%

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

   1. mul-1-neg 78.2%

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

10. Simplified 78.2%

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

11. Taylor expanded in eh around 0 42.5%

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

    1. mul-1-neg 42.5%

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

13. Simplified 42.5%

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

14. Final simplification 42.5%

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

Reproduce

herbie shell --seed 2023297 
(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)))))))