Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 27.4s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

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 99.8%

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

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs(((cos(t_1) * (ew * sin(t))) + (sin(t_1) * (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
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs(((cos(t_1) * (ew * sin(t))) + (sin(t_1) * (eh * cos(t)))))
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(((Math.cos(t_1) * (ew * Math.sin(t))) + (Math.sin(t_1) * (eh * Math.cos(t)))));
}

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = abs(((cos(t_1) * (ew * sin(t))) + (sin(t_1) * (eh * cos(t)))));
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[Cos[t$95$1], $MachinePrecision] * N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t$95$1], $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. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

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 (tan t)) ew)))))))

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 / tan(t)) / ew))))));
}

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 / Math.tan(t)) / ew))))));
}

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 / math.tan(t)) / ew))))))

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(Float64(eh / tan(t)) / ew))))))
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 / tan(t)) / ew))))));
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[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $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. 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. Step-by-step derivation

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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)) * math.cos(math.atan(((eh / t) / ew))))))

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)) * cos(atan(Float64(Float64(eh / t) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((sin(atan(((eh / ew) / tan(t)))) * (eh * cos(t))) + ((ew * sin(t)) * cos(atan(((eh / t) / ew))))));
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[Cos[N[ArcTan[N[(N[(eh / t), $MachinePrecision] / ew), $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 98.8%

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

   1. associate-/r* 88.6%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 5: 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 99.8%

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 85.4%

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

7. Taylor expanded in ew around inf 98.5%

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

8. Final simplification 98.5%

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

Alternative 6: 94.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((ew * sin(t)) + ((eh * cos(t)) * sin(atan(((-0.3333333333333333 * ((t * eh) / ew)) + (eh / (ew * t))))))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((ew * sin(t)) + ((eh * cos(t)) * sin(atan((((-0.3333333333333333d0) * ((t * eh) / ew)) + (eh / (ew * t))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 98.5%

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

7. Taylor expanded in t around 0 95.6%

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

8. Final simplification 95.6%

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

Alternative 7: 89.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 98.5%

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

7. Taylor expanded in t around 0 88.6%

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

   1. associate-/r* 88.6%

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

9. Simplified 88.6%

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

10. Final simplification 88.6%

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

Alternative 8: 79.0% 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 80.5%

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

   1. add-sqr-sqrt 46.2%

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

   2. pow2 46.2%

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

   3. associate-/r* 46.2%

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

   4. cos-atan 55.5%

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

   5. hypot-1-def 53.4%

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

   6. un-div-inv 53.4%

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

   7. *-commutative 53.4%

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

4. Applied egg-rr 53.4%

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

5. Taylor expanded in eh around 0 45.3%

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

   1. unpow2 45.3%

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

   2. add-sqr-sqrt 80.2%

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

   3. *-commutative 80.2%

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

7. Applied egg-rr 80.2%

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

8. Final simplification 80.2%

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

Alternative 9: 76.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((ew * sin(t)) + (eh * sin(atan(((-0.3333333333333333 * ((t * eh) / ew)) + (eh / (ew * t))))))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((ew * sin(t)) + (eh * sin(atan((((-0.3333333333333333d0) * ((t * eh) / ew)) + (eh / (ew * t))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 98.5%

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

7. Taylor expanded in t around 0 80.2%

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

8. Taylor expanded in t around 0 78.9%

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

9. Final simplification 78.9%

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

Alternative 10: 77.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l/ 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in eh around 0 98.5%

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

7. Taylor expanded in t around 0 80.2%

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

8. Taylor expanded in t around 0 78.8%

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

9. Final simplification 78.8%

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

Reproduce

herbie shell --seed 2023279 
(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))))))))