Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 36.4s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   ew
   (/ (sin t) (hypot 1.0 (* (/ eh ew) (/ 1.0 (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) * (1.0 / 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(Float64(eh / ew) * Float64(1.0 / 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[(N[(eh / ew), $MachinePrecision] * N[(1.0 / 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 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-def 99.8%

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

   3. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 99.8%

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

   2. expm1-udef 89.1%

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

   3. cos-atan 89.1%

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

   4. un-div-inv 89.1%

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

   5. hypot-1-def 89.1%

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

   6. associate-/r* 89.1%

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

5. Applied egg-rr 89.1%

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

   1. expm1-def 99.8%

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

   2. expm1-log1p 99.8%

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

7. Simplified 99.8%

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

   1. associate-/r* 99.8%

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

   2. div-inv 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

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 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-def 99.8%

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

   3. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 99.8%

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

   2. expm1-udef 89.1%

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

   3. cos-atan 89.1%

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

   4. un-div-inv 89.1%

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

   5. hypot-1-def 89.1%

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

   6. associate-/r* 89.1%

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

5. Applied egg-rr 89.1%

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

   1. expm1-def 99.8%

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

   2. expm1-log1p 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

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 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
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-/r* 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 4: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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))) + ((ew * Math.sin(t)) * (1.0 / Math.hypot(1.0, (eh / (ew * t)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 99.0%

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

   1. cos-atan 99.0%

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

   2. hypot-1-def 99.0%

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

4. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

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 99.8%

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

   1. associate-*l* 99.8%

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

   2. fma-def 99.8%

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

   3. associate-*l* 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 99.8%

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

   2. expm1-udef 89.1%

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

   3. cos-atan 89.1%

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

   4. un-div-inv 89.1%

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

   5. hypot-1-def 89.1%

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

   6. associate-/r* 89.1%

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

5. Applied egg-rr 89.1%

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

   1. expm1-def 99.8%

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

   2. expm1-log1p 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in eh around 0 98.4%

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

9. Final simplification 98.4%

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

Alternative 6: 79.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. 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 81.5%

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

   1. associate-*l* 81.5%

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

   2. div-inv 81.5%

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

   3. div-inv 81.5%

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

   4. associate-*r* 81.5%

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

   5. frac-times 81.5%

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

   6. metadata-eval 81.5%

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

   7. div-inv 81.5%

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

   8. cos-atan 81.5%

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

   9. hypot-1-def 81.5%

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

   10. div-inv 81.5%

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

   11. clear-num 81.4%

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

   12. un-div-inv 81.4%

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

4. Applied egg-rr 81.4%

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

5. Taylor expanded in ew around inf 80.9%

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

6. Final simplification 80.9%

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

Alternative 7: 35.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 81.5%

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

   1. associate-*l* 81.5%

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

   2. div-inv 81.5%

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

   3. div-inv 81.5%

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

   4. associate-*r* 81.5%

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

   5. frac-times 81.5%

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

   6. metadata-eval 81.5%

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

   7. div-inv 81.5%

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

   8. cos-atan 81.5%

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

   9. hypot-1-def 81.5%

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

   10. div-inv 81.5%

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

   11. clear-num 81.4%

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

   12. un-div-inv 81.4%

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

4. Applied egg-rr 81.4%

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

5. Taylor expanded in t around 0 37.9%

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

   1. associate-/l* 37.4%

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

   2. unpow2 37.4%

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

   3. unpow2 37.4%

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

7. Simplified 37.4%

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

8. Taylor expanded in ew around 0 37.9%

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

   1. unpow2 37.9%

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

   2. associate-*r/ 37.4%

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

   3. unpow2 37.4%

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

   4. associate-/l* 37.9%

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

10. Simplified 37.9%

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

11. Final simplification 37.9%

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

Alternative 8: 35.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 81.5%

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

   1. associate-*l* 81.5%

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

   2. div-inv 81.5%

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

   3. div-inv 81.5%

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

   4. associate-*r* 81.5%

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

   5. frac-times 81.5%

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

   6. metadata-eval 81.5%

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

   7. div-inv 81.5%

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

   8. cos-atan 81.5%

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

   9. hypot-1-def 81.5%

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

   10. div-inv 81.5%

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

   11. clear-num 81.4%

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

   12. un-div-inv 81.4%

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

4. Applied egg-rr 81.4%

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

5. Taylor expanded in t around 0 37.9%

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

   1. associate-/l* 37.4%

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

   2. unpow2 37.4%

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

   3. unpow2 37.4%

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

7. Simplified 37.4%

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

8. Taylor expanded in t around 0 37.4%

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

9. Final simplification 37.4%

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

Reproduce

herbie shell --seed 2023275 
(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))))))))