Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 25.0s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. cos-atan 99.9%

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

   2. hypot-1-def 99.9%

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

   3. associate-/r* 99.9%

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

3. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in t around 0 99.7%

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

3. Final simplification 99.7%

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

Alternative 5: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. cos-atan 99.9%

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

   2. hypot-1-def 99.9%

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

   3. associate-/r* 99.9%

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

3. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eh around 0 98.7%

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

7. Final simplification 98.7%

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

Alternative 6: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -7 \cdot 10^{+61} \lor \neg \left(eh \leq 5.1 \cdot 10^{+182}\right):\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -7e+61) (not (<= eh 5.1e+182)))
   (fabs (* (cos t) (* eh (sin (atan (/ eh (* ew (tan t))))))))
   (fabs (+ (* ew (sin t)) (* eh (sin (atan (/ (/ eh ew) (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7e+61) || !(eh <= 5.1e+182)) {
		tmp = fabs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	} else {
		tmp = fabs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	}
	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) :: tmp
    if ((eh <= (-7d+61)) .or. (.not. (eh <= 5.1d+182))) then
        tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))))
    else
        tmp = abs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7e+61) || !(eh <= 5.1e+182)) {
		tmp = Math.abs((Math.cos(t) * (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	} else {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -7e+61) or not (eh <= 5.1e+182):
		tmp = math.fabs((math.cos(t) * (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	else:
		tmp = math.fabs(((ew * math.sin(t)) + (eh * math.sin(math.atan(((eh / ew) / math.tan(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -7e+61) || !(eh <= 5.1e+182))
		tmp = abs(Float64(cos(t) * Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	else
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -7e+61) || ~((eh <= 5.1e+182)))
		tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	else
		tmp = abs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -7e+61], N[Not[LessEqual[eh, 5.1e+182]], $MachinePrecision]], N[Abs[N[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.00000000000000036e61 or 5.10000000000000009e182 < eh

   1. Initial program 99.8%

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

      1. fma-def 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 80.5%

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

   5. Taylor expanded in ew around 0 93.1%

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

   if -7.00000000000000036e61 < eh < 5.10000000000000009e182

   1. Initial program 99.9%

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

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. associate-/r* 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eh around 0 98.3%

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

   7. Taylor expanded in t around 0 91.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7 \cdot 10^{+61} \lor \neg \left(eh \leq 5.1 \cdot 10^{+182}\right):\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \]

Alternative 7: 89.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.45 \cdot 10^{+105}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -2.45e+105)
   (fabs (* (cos t) (* eh (sin (atan (/ eh (* ew (tan t))))))))
   (fabs (+ (* ew (sin t)) (* (* eh (cos t)) (sin (atan (/ eh (* ew t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2.45e+105) {
		tmp = fabs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	} else {
		tmp = fabs(((ew * sin(t)) + ((eh * cos(t)) * sin(atan((eh / (ew * t)))))));
	}
	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) :: tmp
    if (eh <= (-2.45d+105)) then
        tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))))
    else
        tmp = abs(((ew * sin(t)) + ((eh * cos(t)) * sin(atan((eh / (ew * t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2.45e+105) {
		tmp = Math.abs((Math.cos(t) * (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	} else {
		tmp = Math.abs(((ew * Math.sin(t)) + ((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (ew * t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -2.45e+105:
		tmp = math.fabs((math.cos(t) * (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	else:
		tmp = math.fabs(((ew * math.sin(t)) + ((eh * math.cos(t)) * math.sin(math.atan((eh / (ew * t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -2.45e+105)
		tmp = abs(Float64(cos(t) * Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	else
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -2.45e+105)
		tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	else
		tmp = abs(((ew * sin(t)) + ((eh * cos(t)) * sin(atan((eh / (ew * t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -2.45e+105], N[Abs[N[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.45e105

   1. Initial program 99.8%

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

      1. fma-def 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 75.9%

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

   5. Taylor expanded in ew around 0 93.7%

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

   if -2.45e105 < eh

   1. Initial program 99.9%

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

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. associate-/r* 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eh around 0 98.5%

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

   7. Taylor expanded in t around 0 92.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.45 \cdot 10^{+105}:\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]

Alternative 8: 85.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.8 \cdot 10^{+62} \lor \neg \left(eh \leq 5.1 \cdot 10^{+182}\right):\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.8e+62) (not (<= eh 5.1e+182)))
   (fabs (* (cos t) (* eh (sin (atan (/ eh (* ew (tan t))))))))
   (fabs (+ (* ew (sin t)) (* eh (sin (atan (/ eh (* ew t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.8e+62) || !(eh <= 5.1e+182)) {
		tmp = fabs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	} else {
		tmp = fabs(((ew * sin(t)) + (eh * sin(atan((eh / (ew * t)))))));
	}
	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) :: tmp
    if ((eh <= (-1.8d+62)) .or. (.not. (eh <= 5.1d+182))) then
        tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))))
    else
        tmp = abs(((ew * sin(t)) + (eh * sin(atan((eh / (ew * t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.8e+62) || !(eh <= 5.1e+182)) {
		tmp = Math.abs((Math.cos(t) * (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	} else {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * Math.sin(Math.atan((eh / (ew * t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.8e+62) or not (eh <= 5.1e+182):
		tmp = math.fabs((math.cos(t) * (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	else:
		tmp = math.fabs(((ew * math.sin(t)) + (eh * math.sin(math.atan((eh / (ew * t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.8e+62) || !(eh <= 5.1e+182))
		tmp = abs(Float64(cos(t) * Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	else
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(eh / Float64(ew * t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -1.8e+62) || ~((eh <= 5.1e+182)))
		tmp = abs((cos(t) * (eh * sin(atan((eh / (ew * tan(t))))))));
	else
		tmp = abs(((ew * sin(t)) + (eh * sin(atan((eh / (ew * t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.8e+62], N[Not[LessEqual[eh, 5.1e+182]], $MachinePrecision]], N[Abs[N[(N[Cos[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.8e62 or 5.10000000000000009e182 < eh

   1. Initial program 99.8%

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

      1. fma-def 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 80.5%

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

   5. Taylor expanded in ew around 0 93.1%

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

   if -1.8e62 < eh < 5.10000000000000009e182

   1. Initial program 99.9%

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

      1. cos-atan 99.9%

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

      2. hypot-1-def 99.9%

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

      3. associate-/r* 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in eh around 0 98.3%

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

   7. Taylor expanded in t around 0 91.5%

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

   8. Taylor expanded in t around 0 90.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.8 \cdot 10^{+62} \lor \neg \left(eh \leq 5.1 \cdot 10^{+182}\right):\\ \;\;\;\;\left|\cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]

Alternative 9: 45.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{if}\;ew \leq -4.3 \cdot 10^{+171} \lor \neg \left(ew \leq 5 \cdot 10^{+194}\right):\\ \;\;\;\;\left|t \cdot \left(ew \cdot \cos t_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* ew (tan t))))))
   (if (or (<= ew -4.3e+171) (not (<= ew 5e+194)))
     (fabs (* t (* ew (cos t_1))))
     (fabs (* eh (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * tan(t))));
	double tmp;
	if ((ew <= -4.3e+171) || !(ew <= 5e+194)) {
		tmp = fabs((t * (ew * cos(t_1))));
	} else {
		tmp = fabs((eh * sin(t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan((eh / (ew * tan(t))))
    if ((ew <= (-4.3d+171)) .or. (.not. (ew <= 5d+194))) then
        tmp = abs((t * (ew * cos(t_1))))
    else
        tmp = abs((eh * sin(t_1)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((eh / (ew * Math.tan(t))));
	double tmp;
	if ((ew <= -4.3e+171) || !(ew <= 5e+194)) {
		tmp = Math.abs((t * (ew * Math.cos(t_1))));
	} else {
		tmp = Math.abs((eh * Math.sin(t_1)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh / (ew * math.tan(t))))
	tmp = 0
	if (ew <= -4.3e+171) or not (ew <= 5e+194):
		tmp = math.fabs((t * (ew * math.cos(t_1))))
	else:
		tmp = math.fabs((eh * math.sin(t_1)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * tan(t))))
	tmp = 0.0
	if ((ew <= -4.3e+171) || !(ew <= 5e+194))
		tmp = abs(Float64(t * Float64(ew * cos(t_1))));
	else
		tmp = abs(Float64(eh * sin(t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh / (ew * tan(t))));
	tmp = 0.0;
	if ((ew <= -4.3e+171) || ~((ew <= 5e+194)))
		tmp = abs((t * (ew * cos(t_1))));
	else
		tmp = abs((eh * sin(t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -4.3e+171], N[Not[LessEqual[ew, 5e+194]], $MachinePrecision]], N[Abs[N[(t * N[(ew * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.30000000000000008e171 or 4.99999999999999989e194 < ew

   1. Initial program 99.9%

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

      1. fma-def 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 52.8%

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

   5. Taylor expanded in t around -inf 39.6%

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

   if -4.30000000000000008e171 < ew < 4.99999999999999989e194

   1. Initial program 99.9%

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

      1. fma-def 99.9%

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

      2. associate-/l/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 69.4%

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

   5. Taylor expanded in t around 0 52.4%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.3 \cdot 10^{+171} \lor \neg \left(ew \leq 5 \cdot 10^{+194}\right):\\ \;\;\;\;\left|t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]

Alternative 10: 78.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. cos-atan 99.9%

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

   2. hypot-1-def 99.9%

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

   3. associate-/r* 99.9%

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

3. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in eh around 0 98.7%

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

7. Taylor expanded in t around 0 83.2%

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

8. Taylor expanded in t around 0 82.0%

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

9. Final simplification 82.0%

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

Alternative 11: 41.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around 0 65.8%

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

5. Taylor expanded in t around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. *-lft-identity 44.1%

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

   3. times-frac 44.1%

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

   4. associate-*l/ 44.1%

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

   5. *-lft-identity 44.1%

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

7. Simplified 44.1%

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

8. Final simplification 44.1%

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

Alternative 12: 41.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around 0 65.8%

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

5. Taylor expanded in t around 0 44.1%

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

6. Final simplification 44.1%

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

Alternative 13: 39.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t} + -0.3333333333333333 \cdot \frac{t \cdot eh}{ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   eh
   (sin (atan (+ (/ eh (* ew t)) (* -0.3333333333333333 (/ (* t eh) ew))))))))

C

double code(double eh, double ew, double t) {
	return fabs((eh * sin(atan(((eh / (ew * t)) + (-0.3333333333333333 * ((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((eh * sin(atan(((eh / (ew * t)) + ((-0.3333333333333333d0) * ((t * eh) / ew)))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((eh * Math.sin(Math.atan(((eh / (ew * t)) + (-0.3333333333333333 * ((t * eh) / ew)))))));
}

Python

def code(eh, ew, t):
	return math.fabs((eh * math.sin(math.atan(((eh / (ew * t)) + (-0.3333333333333333 * ((t * eh) / ew)))))))

Julia

function code(eh, ew, t)
	return abs(Float64(eh * sin(atan(Float64(Float64(eh / Float64(ew * t)) + Float64(-0.3333333333333333 * Float64(Float64(t * eh) / ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((eh * sin(atan(((eh / (ew * t)) + (-0.3333333333333333 * ((t * eh) / ew)))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around 0 65.8%

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

5. Taylor expanded in t around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. *-lft-identity 44.1%

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

   3. times-frac 44.1%

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

   4. associate-*l/ 44.1%

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

   5. *-lft-identity 44.1%

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

7. Simplified 44.1%

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

8. Taylor expanded in t around 0 43.6%

   \[\leadsto \left|eh \cdot \sin \tan^{-1} \color{blue}{\left(-0.3333333333333333 \cdot \frac{t \cdot eh}{ew} + \frac{eh}{t \cdot ew}\right)}\right| \]

9. Final simplification 43.6%

   \[\leadsto \left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t} + -0.3333333333333333 \cdot \frac{t \cdot eh}{ew}\right)\right| \]

Alternative 14: 39.9% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* eh (sin (atan (* (/ eh ew) (/ 1.0 t)))))))

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((eh * sin(atan(((eh / ew) * (1.0d0 / t))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around 0 65.8%

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

5. Taylor expanded in t around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. *-lft-identity 44.1%

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

   3. times-frac 44.1%

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

   4. associate-*l/ 44.1%

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

   5. *-lft-identity 44.1%

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

7. Simplified 44.1%

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

8. Taylor expanded in t around 0 42.6%

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

   1. *-un-lft-identity 42.6%

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

   2. times-frac 42.7%

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

10. Applied egg-rr 42.7%

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

11. Final simplification 42.7%

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

Alternative 15: 39.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around 0 65.8%

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

5. Taylor expanded in t around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. *-lft-identity 44.1%

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

   3. times-frac 44.1%

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

   4. associate-*l/ 44.1%

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

   5. *-lft-identity 44.1%

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

7. Simplified 44.1%

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

8. Taylor expanded in t around 0 42.6%

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

9. Final simplification 42.6%

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

Alternative 16: 17.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around 0 65.8%

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

5. Taylor expanded in t around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. *-lft-identity 44.1%

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

   3. times-frac 44.1%

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

   4. associate-*l/ 44.1%

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

   5. *-lft-identity 44.1%

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

7. Simplified 44.1%

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

8. Taylor expanded in t around 0 42.6%

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

   1. sin-atan 13.9%

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

   2. associate-/r* 13.2%

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

   3. associate-/l/ 13.4%

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

   4. hypot-1-def 19.1%

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

10. Applied egg-rr 19.1%

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

11. Final simplification 19.1%

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

Alternative 17: 20.9% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around 0 65.8%

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

5. Taylor expanded in t around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. *-lft-identity 44.1%

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

   3. times-frac 44.1%

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

   4. associate-*l/ 44.1%

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

   5. *-lft-identity 44.1%

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

7. Simplified 44.1%

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

8. Taylor expanded in t around 0 42.6%

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

   1. *-commutative 42.6%

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

   2. sin-atan 13.9%

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

   3. associate-*l/ 13.5%

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

   4. hypot-1-def 16.1%

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

10. Applied egg-rr 16.1%

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

    1. associate-/l* 21.2%

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

    2. associate-/r/ 22.0%

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

    3. associate-/l/ 22.1%

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

    4. associate-*r* 21.0%

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

    5. *-commutative 21.0%

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

    6. *-commutative 21.0%

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

12. Simplified 21.0%

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

13. Final simplification 21.0%

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

Alternative 18: 21.4% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t)))) (fabs (/ eh (/ (hypot 1.0 t_1) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	return fabs((eh / (hypot(1.0, t_1) / t_1)));
}

Java

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

Python

def code(eh, ew, t):
	t_1 = eh / (ew * t)
	return math.fabs((eh / (math.hypot(1.0, t_1) / t_1)))

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	return abs(Float64(eh / Float64(hypot(1.0, t_1) / t_1)))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = eh / (ew * t);
	tmp = abs((eh / (hypot(1.0, t_1) / t_1)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-def 99.9%

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

   2. associate-/l/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around 0 65.8%

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

5. Taylor expanded in t around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. *-lft-identity 44.1%

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

   3. times-frac 44.1%

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

   4. associate-*l/ 44.1%

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

   5. *-lft-identity 44.1%

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

7. Simplified 44.1%

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

8. Taylor expanded in t around 0 42.6%

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

   1. *-commutative 42.6%

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

   2. sin-atan 13.9%

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

   3. associate-*l/ 13.5%

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

   4. hypot-1-def 16.1%

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

10. Applied egg-rr 16.1%

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

    1. *-commutative 16.1%

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

    2. associate-/l* 22.0%

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

12. Simplified 22.0%

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

13. Final simplification 22.0%

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

Reproduce

herbie shell --seed 2023257 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))