Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 33.0s

Alternatives: 13

Speedup: 0.9×

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

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

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(Float64(eh / ew) / tan(t)))
	return abs(fma(ew, Float64(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[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 88.2%

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

   3. cos-atan 88.2%

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

   4. un-div-inv 88.2%

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

   5. hypot-1-def 88.2%

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

   6. associate-/l/ 88.2%

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

   7. *-commutative 88.2%

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

5. Applied egg-rr 88.2%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. associate-/l/ 99.7%

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

   4. *-commutative 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 88.2%

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

   3. cos-atan 88.2%

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

   4. un-div-inv 88.2%

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

   5. hypot-1-def 88.2%

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

   6. associate-/l/ 88.2%

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

   7. *-commutative 88.2%

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

5. Applied egg-rr 88.2%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 98.5%

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

   1. *-commutative 98.5%

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

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 5: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 88.2%

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

   3. cos-atan 88.2%

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

   4. un-div-inv 88.2%

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

   5. hypot-1-def 88.2%

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

   6. associate-/l/ 88.2%

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

   7. *-commutative 88.2%

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

5. Applied egg-rr 88.2%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in eh around 0 97.8%

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

9. Final simplification 97.8%

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

Alternative 6: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. cos-atan 99.7%

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

   2. hypot-1-def 99.7%

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

   3. associate-/l/ 99.7%

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

   4. *-commutative 99.7%

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

3. Applied egg-rr 99.7%

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

4. Taylor expanded in eh around 0 97.8%

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

5. Final simplification 97.8%

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

Alternative 7: 90.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;ew \leq -1.16 \cdot 10^{-207} \lor \neg \left(ew \leq 2.3 \cdot 10^{-144}\right):\\ \;\;\;\;\left|ew \cdot \sin t + t_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot t_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (or (<= ew -1.16e-207) (not (<= ew 2.3e-144)))
     (fabs (+ (* ew (sin t)) (* t_1 (sin (atan (/ eh (* ew t)))))))
     (fabs (* (sin (atan (/ (/ eh ew) (tan t)))) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if ((ew <= -1.16e-207) || !(ew <= 2.3e-144)) {
		tmp = fabs(((ew * sin(t)) + (t_1 * sin(atan((eh / (ew * t)))))));
	} else {
		tmp = fabs((sin(atan(((eh / ew) / tan(t)))) * 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 = eh * cos(t)
    if ((ew <= (-1.16d-207)) .or. (.not. (ew <= 2.3d-144))) then
        tmp = abs(((ew * sin(t)) + (t_1 * sin(atan((eh / (ew * t)))))))
    else
        tmp = abs((sin(atan(((eh / ew) / tan(t)))) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double tmp;
	if ((ew <= -1.16e-207) || !(ew <= 2.3e-144)) {
		tmp = Math.abs(((ew * Math.sin(t)) + (t_1 * Math.sin(Math.atan((eh / (ew * t)))))));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((eh / ew) / Math.tan(t)))) * t_1));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	tmp = 0
	if (ew <= -1.16e-207) or not (ew <= 2.3e-144):
		tmp = math.fabs(((ew * math.sin(t)) + (t_1 * math.sin(math.atan((eh / (ew * t)))))))
	else:
		tmp = math.fabs((math.sin(math.atan(((eh / ew) / math.tan(t)))) * t_1))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if ((ew <= -1.16e-207) || !(ew <= 2.3e-144))
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(t_1 * sin(atan(Float64(eh / Float64(ew * t)))))));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / ew) / tan(t)))) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	tmp = 0.0;
	if ((ew <= -1.16e-207) || ~((ew <= 2.3e-144)))
		tmp = abs(((ew * sin(t)) + (t_1 * sin(atan((eh / (ew * t)))))));
	else
		tmp = abs((sin(atan(((eh / ew) / tan(t)))) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -1.16e-207], N[Not[LessEqual[ew, 2.3e-144]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.16e-207 or 2.3e-144 < ew

   1. Initial program 99.8%

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

      1. cos-atan 99.8%

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

      2. hypot-1-def 99.8%

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

      3. associate-/l/ 99.8%

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

      4. *-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in eh around 0 97.5%

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

   5. Taylor expanded in t around 0 88.9%

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

   if -1.16e-207 < ew < 2.3e-144

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. fma-def 99.7%

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

      3. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. expm1-udef 99.7%

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

      3. cos-atan 99.7%

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

      4. un-div-inv 99.7%

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

      5. hypot-1-def 99.7%

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

      6. associate-/l/ 99.7%

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

      7. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 69.8%

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

      1. associate-/l* 68.3%

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

      2. unpow2 68.3%

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

   10. Simplified 68.3%

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

   11. Taylor expanded in ew around 0 98.1%

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

       1. associate-*r* 98.1%

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

       2. associate-/r* 98.1%

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

   13. Simplified 98.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.16 \cdot 10^{-207} \lor \neg \left(ew \leq 2.3 \cdot 10^{-144}\right):\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \end{array} \]

Alternative 8: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;ew \leq -4.2 \cdot 10^{-144} \lor \neg \left(ew \leq 2.7 \cdot 10^{-146}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \left(eh \cdot \cos t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ (/ eh ew) (tan t))))))
   (if (or (<= ew -4.2e-144) (not (<= ew 2.7e-146)))
     (fabs (+ (* ew (sin t)) (* eh t_1)))
     (fabs (* t_1 (* eh (cos t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan(((eh / ew) / tan(t))));
	double tmp;
	if ((ew <= -4.2e-144) || !(ew <= 2.7e-146)) {
		tmp = fabs(((ew * sin(t)) + (eh * t_1)));
	} else {
		tmp = fabs((t_1 * (eh * cos(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) :: t_1
    real(8) :: tmp
    t_1 = sin(atan(((eh / ew) / tan(t))))
    if ((ew <= (-4.2d-144)) .or. (.not. (ew <= 2.7d-146))) then
        tmp = abs(((ew * sin(t)) + (eh * t_1)))
    else
        tmp = abs((t_1 * (eh * cos(t))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan(((eh / ew) / Math.tan(t))));
	double tmp;
	if ((ew <= -4.2e-144) || !(ew <= 2.7e-146)) {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * t_1)));
	} else {
		tmp = Math.abs((t_1 * (eh * Math.cos(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan(((eh / ew) / math.tan(t))))
	tmp = 0
	if (ew <= -4.2e-144) or not (ew <= 2.7e-146):
		tmp = math.fabs(((ew * math.sin(t)) + (eh * t_1)))
	else:
		tmp = math.fabs((t_1 * (eh * math.cos(t))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(Float64(eh / ew) / tan(t))))
	tmp = 0.0
	if ((ew <= -4.2e-144) || !(ew <= 2.7e-146))
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * t_1)));
	else
		tmp = abs(Float64(t_1 * Float64(eh * cos(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan(((eh / ew) / tan(t))));
	tmp = 0.0;
	if ((ew <= -4.2e-144) || ~((ew <= 2.7e-146)))
		tmp = abs(((ew * sin(t)) + (eh * t_1)));
	else
		tmp = abs((t_1 * (eh * cos(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -4.2e-144], N[Not[LessEqual[ew, 2.7e-146]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.2000000000000002e-144 or 2.69999999999999995e-146 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0 85.7%

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

      1. add-sqr-sqrt 50.3%

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

      2. pow2 50.3%

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

   4. Applied egg-rr 56.8%

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

      1. unpow2 56.8%

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

      2. add-sqr-sqrt 85.6%

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

      3. clear-num 85.5%

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

      4. associate-/r* 85.5%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in eh around 0 84.8%

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

   if -4.2000000000000002e-144 < ew < 2.69999999999999995e-146

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. fma-def 99.7%

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

      3. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 99.6%

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

      3. cos-atan 99.6%

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

      4. un-div-inv 99.6%

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

      5. hypot-1-def 99.6%

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

      6. associate-/l/ 99.6%

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

      7. *-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 70.3%

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

      1. associate-/l* 69.0%

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

      2. unpow2 69.0%

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

   10. Simplified 69.0%

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

   11. Taylor expanded in ew around 0 95.9%

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

       1. associate-*r* 95.9%

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

       2. associate-/r* 95.9%

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

   13. Simplified 95.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.2 \cdot 10^{-144} \lor \neg \left(ew \leq 2.7 \cdot 10^{-146}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \end{array} \]

Alternative 9: 60.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 88.2%

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

   3. cos-atan 88.2%

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

   4. un-div-inv 88.2%

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

   5. hypot-1-def 88.2%

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

   6. associate-/l/ 88.2%

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

   7. *-commutative 88.2%

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

5. Applied egg-rr 88.2%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 48.4%

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

   1. associate-/l* 48.0%

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

   2. unpow2 48.0%

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

10. Simplified 48.0%

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

11. Taylor expanded in ew around 0 63.6%

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

    1. associate-*r* 63.6%

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

    2. associate-/r* 63.6%

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

13. Simplified 63.6%

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

14. Final simplification 63.6%

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

Alternative 10: 41.5% 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 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 88.2%

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

   3. cos-atan 88.2%

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

   4. un-div-inv 88.2%

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

   5. hypot-1-def 88.2%

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

   6. associate-/l/ 88.2%

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

   7. *-commutative 88.2%

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

5. Applied egg-rr 88.2%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 48.4%

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

   1. associate-/l* 48.0%

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

   2. unpow2 48.0%

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

10. Simplified 48.0%

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

11. Taylor expanded in t around 0 41.2%

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

12. Final simplification 41.2%

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

Alternative 11: 3.8% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 88.2%

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

   3. cos-atan 88.2%

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

   4. un-div-inv 88.2%

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

   5. hypot-1-def 88.2%

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

   6. associate-/l/ 88.2%

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

   7. *-commutative 88.2%

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

5. Applied egg-rr 88.2%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 48.4%

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

   1. associate-/l* 48.0%

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

   2. unpow2 48.0%

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

10. Simplified 48.0%

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

11. Taylor expanded in ew around inf 4.0%

    \[\leadsto \left|\color{blue}{\frac{{ew}^{2} \cdot {t}^{2}}{eh}}\right| \]
12. Step-by-step derivation

    1. associate-/l* 3.9%

       \[\leadsto \left|\color{blue}{\frac{{ew}^{2}}{\frac{eh}{{t}^{2}}}}\right| \]

    2. unpow2 3.9%

       \[\leadsto \left|\frac{\color{blue}{ew \cdot ew}}{\frac{eh}{{t}^{2}}}\right| \]

    3. unpow2 3.9%

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

13. Simplified 3.9%

    \[\leadsto \left|\color{blue}{\frac{ew \cdot ew}{\frac{eh}{t \cdot t}}}\right| \]
14. Step-by-step derivation

    1. div-inv 3.9%

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

    2. clear-num 3.9%

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

15. Applied egg-rr 3.9%

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

16. Final simplification 3.9%

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

Alternative 12: 3.9% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 88.2%

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

   3. cos-atan 88.2%

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

   4. un-div-inv 88.2%

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

   5. hypot-1-def 88.2%

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

   6. associate-/l/ 88.2%

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

   7. *-commutative 88.2%

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

5. Applied egg-rr 88.2%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 48.4%

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

   1. associate-/l* 48.0%

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

   2. unpow2 48.0%

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

10. Simplified 48.0%

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

11. Taylor expanded in ew around inf 4.0%

    \[\leadsto \left|\color{blue}{\frac{{ew}^{2} \cdot {t}^{2}}{eh}}\right| \]
12. Step-by-step derivation

    1. associate-/l* 3.9%

       \[\leadsto \left|\color{blue}{\frac{{ew}^{2}}{\frac{eh}{{t}^{2}}}}\right| \]

    2. unpow2 3.9%

       \[\leadsto \left|\frac{\color{blue}{ew \cdot ew}}{\frac{eh}{{t}^{2}}}\right| \]

    3. unpow2 3.9%

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

13. Simplified 3.9%

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

14. Taylor expanded in ew around 0 4.0%

    \[\leadsto \left|\color{blue}{\frac{{ew}^{2} \cdot {t}^{2}}{eh}}\right| \]
15. Step-by-step derivation

    1. *-commutative 4.0%

       \[\leadsto \left|\frac{\color{blue}{{t}^{2} \cdot {ew}^{2}}}{eh}\right| \]

    2. associate-*r/ 3.9%

       \[\leadsto \left|\color{blue}{{t}^{2} \cdot \frac{{ew}^{2}}{eh}}\right| \]

    3. unpow2 3.9%

       \[\leadsto \left|\color{blue}{\left(t \cdot t\right)} \cdot \frac{{ew}^{2}}{eh}\right| \]

    4. unpow2 3.9%

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

    5. associate-/l* 4.0%

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

16. Simplified 4.0%

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

17. Final simplification 4.0%

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

Alternative 13: 4.4% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. fma-def 99.7%

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

   3. associate-*l* 99.7%

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

3. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 88.2%

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

   3. cos-atan 88.2%

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

   4. un-div-inv 88.2%

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

   5. hypot-1-def 88.2%

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

   6. associate-/l/ 88.2%

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

   7. *-commutative 88.2%

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

5. Applied egg-rr 88.2%

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

   1. expm1-def 99.7%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in t around 0 48.4%

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

   1. associate-/l* 48.0%

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

   2. unpow2 48.0%

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

10. Simplified 48.0%

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

11. Taylor expanded in ew around inf 4.0%

    \[\leadsto \left|\color{blue}{\frac{{ew}^{2} \cdot {t}^{2}}{eh}}\right| \]
12. Step-by-step derivation

    1. associate-/l* 3.9%

       \[\leadsto \left|\color{blue}{\frac{{ew}^{2}}{\frac{eh}{{t}^{2}}}}\right| \]

    2. unpow2 3.9%

       \[\leadsto \left|\frac{\color{blue}{ew \cdot ew}}{\frac{eh}{{t}^{2}}}\right| \]

    3. unpow2 3.9%

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

13. Simplified 3.9%

    \[\leadsto \left|\color{blue}{\frac{ew \cdot ew}{\frac{eh}{t \cdot t}}}\right| \]
14. Step-by-step derivation

    1. associate-/r/ 3.9%

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

    2. associate-*r* 4.5%

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

15. Applied egg-rr 4.5%

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

16. Final simplification 4.5%

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

Reproduce

herbie shell --seed 2023283 
(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))))))))