Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.0s

Alternatives: 10

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, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ \left|\frac{ew \cdot \sin t}{\mathsf{hypot}\left(1, t\_1\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t))))
   (fabs
    (+
     (/ (* ew (sin t)) (hypot 1.0 t_1))
     (* (* eh (cos t)) (sin (atan t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	return fabs((((ew * sin(t)) / hypot(1.0, t_1)) + ((eh * cos(t)) * sin(atan(t_1)))));
}

Java

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

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / math.tan(t)
	return math.fabs((((ew * math.sin(t)) / math.hypot(1.0, t_1)) + ((eh * math.cos(t)) * math.sin(math.atan(t_1)))))

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(Float64(Float64(Float64(ew * sin(t)) / hypot(1.0, t_1)) + Float64(Float64(eh * cos(t)) * sin(atan(t_1)))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / tan(t);
	tmp = abs((((ew * sin(t)) / hypot(1.0, t_1)) + ((eh * cos(t)) * sin(atan(t_1)))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l/ 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l/ 99.8%

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

   2. cos-atan 99.8%

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

   3. un-div-inv 99.8%

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

   4. hypot-1-def 99.8%

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

   5. associate-/l/ 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around inf 98.0%

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

Alternative 4: 87.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{-44} \lor \neg \left(ew \leq 5 \cdot 10^{-113}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.9e-44) (not (<= ew 5e-113)))
   (fabs (+ (* ew (sin t)) (* eh (sin (atan (/ (/ eh ew) (tan t)))))))
   (fabs (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.9e-44) || !(ew <= 5e-113)) {
		tmp = fabs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = fabs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.9d-44)) .or. (.not. (ew <= 5d-113))) then
        tmp = abs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))))
    else
        tmp = abs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.9e-44) || !(ew <= 5e-113)) {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.9e-44) or not (ew <= 5e-113):
		tmp = math.fabs(((ew * math.sin(t)) + (eh * math.sin(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = math.fabs((eh * (math.cos(t) * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.9e-44) || !(ew <= 5e-113))
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.9e-44) || ~((ew <= 5e-113)))
		tmp = abs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	else
		tmp = abs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.9e-44], N[Not[LessEqual[ew, 5e-113]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.9e-44 or 4.9999999999999997e-113 < ew

   1. Initial program 99.8%

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

      1. associate-/l/ 99.8%

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

      2. cos-atan 99.8%

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

      3. un-div-inv 99.8%

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

      4. hypot-1-def 99.8%

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

      5. associate-/l/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf 97.2%

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

   6. Taylor expanded in t around 0 88.6%

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

   if -1.9e-44 < ew < 4.9999999999999997e-113

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 94.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{-44} \lor \neg \left(ew \leq 5 \cdot 10^{-113}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{-14} \lor \neg \left(eh \leq 1.5 \cdot 10^{-75}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.3e-14) (not (<= eh 1.5e-75)))
   (fabs (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t))))))))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.3e-14) || !(eh <= 1.5e-75)) {
		tmp = fabs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-3.3d-14)) .or. (.not. (eh <= 1.5d-75))) then
        tmp = abs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))))
    else
        tmp = abs((ew * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.3e-14) || !(eh <= 1.5e-75)) {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	} else {
		tmp = Math.abs((ew * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -3.3e-14) or not (eh <= 1.5e-75):
		tmp = math.fabs((eh * (math.cos(t) * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.3e-14) || !(eh <= 1.5e-75))
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -3.3e-14) || ~((eh <= 1.5e-75)))
		tmp = abs((eh * (cos(t) * sin(atan((eh / (ew * tan(t))))))));
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.3e-14], N[Not[LessEqual[eh, 1.5e-75]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.2999999999999998e-14 or 1.4999999999999999e-75 < eh

   1. Initial program 99.7%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 83.0%

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

   if -3.2999999999999998e-14 < eh < 1.4999999999999999e-75

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 45.3%

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

      2. fabs-sqr 45.3%

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

      3. add-sqr-sqrt 46.8%

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

      4. expm1-log1p-u 45.0%

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

      5. expm1-undefine 21.6%

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

   6. Applied egg-rr 21.6%

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

      1. expm1-define 45.0%

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

      2. fma-define 45.0%

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

      3. +-commutative 45.0%

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

      4. fma-define 45.0%

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

      5. associate-/r* 45.0%

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

      6. associate-*r/ 44.9%

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

      7. associate-/r* 44.9%

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

   8. Simplified 44.9%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in ew around inf 77.8%

       \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{-14} \lor \neg \left(eh \leq 1.5 \cdot 10^{-75}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ \mathbf{if}\;eh \leq -3.6 \cdot 10^{-14}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot t\_1\right)\right|\\ \mathbf{elif}\;eh \leq 4.9 \cdot 10^{-64}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ eh (* ew (tan t)))))))
   (if (<= eh -3.6e-14)
     (fabs (* eh (* (cos t) t_1)))
     (if (<= eh 4.9e-64)
       (fabs (* ew (sin t)))
       (fabs (* (* eh (cos t)) t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan((eh / (ew * tan(t)))));
	double tmp;
	if (eh <= -3.6e-14) {
		tmp = fabs((eh * (cos(t) * t_1)));
	} else if (eh <= 4.9e-64) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(((eh * cos(t)) * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(atan((eh / (ew * tan(t)))))
    if (eh <= (-3.6d-14)) then
        tmp = abs((eh * (cos(t) * t_1)))
    else if (eh <= 4.9d-64) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs(((eh * cos(t)) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(Math.atan((eh / (ew * Math.tan(t)))));
	double tmp;
	if (eh <= -3.6e-14) {
		tmp = Math.abs((eh * (Math.cos(t) * t_1)));
	} else if (eh <= 4.9e-64) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs(((eh * Math.cos(t)) * t_1));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(math.atan((eh / (ew * math.tan(t)))))
	tmp = 0
	if eh <= -3.6e-14:
		tmp = math.fabs((eh * (math.cos(t) * t_1)))
	elif eh <= 4.9e-64:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(((eh * math.cos(t)) * t_1))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(eh / Float64(ew * tan(t)))))
	tmp = 0.0
	if (eh <= -3.6e-14)
		tmp = abs(Float64(eh * Float64(cos(t) * t_1)));
	elseif (eh <= 4.9e-64)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(Float64(eh * cos(t)) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(atan((eh / (ew * tan(t)))));
	tmp = 0.0;
	if (eh <= -3.6e-14)
		tmp = abs((eh * (cos(t) * t_1)));
	elseif (eh <= 4.9e-64)
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(((eh * cos(t)) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.6e-14], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.9e-64], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -3.5999999999999998e-14

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 83.5%

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

   if -3.5999999999999998e-14 < eh < 4.9000000000000002e-64

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 45.3%

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

      2. fabs-sqr 45.3%

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

      3. add-sqr-sqrt 46.8%

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

      4. expm1-log1p-u 45.0%

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

      5. expm1-undefine 21.6%

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

   6. Applied egg-rr 21.6%

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

      1. expm1-define 45.0%

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

      2. fma-define 45.0%

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

      3. +-commutative 45.0%

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

      4. fma-define 45.0%

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

      5. associate-/r* 45.0%

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

      6. associate-*r/ 44.9%

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

      7. associate-/r* 44.9%

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

   8. Simplified 44.9%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in ew around inf 77.8%

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

   if 4.9000000000000002e-64 < eh

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

      2. associate-/l/ 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in ew around 0 82.6%

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

      1. associate-*r* 82.6%

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

      2. *-commutative 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.6 \cdot 10^{-14}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 4.9 \cdot 10^{-64}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.9 \cdot 10^{-14} \lor \neg \left(eh \leq 4.8 \cdot 10^{+116}\right):\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.9e-14) (not (<= eh 4.8e+116)))
   (fabs eh)
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.9e-14) || !(eh <= 4.8e+116)) {
		tmp = fabs(eh);
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-3.9d-14)) .or. (.not. (eh <= 4.8d+116))) then
        tmp = abs(eh)
    else
        tmp = abs((ew * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.9e-14) || !(eh <= 4.8e+116)) {
		tmp = Math.abs(eh);
	} else {
		tmp = Math.abs((ew * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -3.9e-14) or not (eh <= 4.8e+116):
		tmp = math.fabs(eh)
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.9e-14) || !(eh <= 4.8e+116))
		tmp = abs(eh);
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -3.9e-14) || ~((eh <= 4.8e+116)))
		tmp = abs(eh);
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.9e-14], N[Not[LessEqual[eh, 4.8e+116]], $MachinePrecision]], N[Abs[eh], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.8999999999999998e-14 or 4.8000000000000001e116 < eh

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 61.3%

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

      1. sin-atan 13.3%

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

      2. hypot-1-def 29.9%

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

      3. clear-num 29.9%

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

   7. Applied egg-rr 29.9%

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

   8. Taylor expanded in eh around inf 61.5%

      \[\leadsto \left|\color{blue}{eh}\right| \]

   if -3.8999999999999998e-14 < eh < 4.8000000000000001e116

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 47.9%

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

      2. fabs-sqr 47.9%

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

      3. add-sqr-sqrt 49.3%

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

      4. expm1-log1p-u 47.1%

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

      5. expm1-undefine 25.9%

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

   6. Applied egg-rr 25.9%

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

      1. expm1-define 47.1%

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

      2. fma-define 47.1%

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

      3. +-commutative 47.1%

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

      4. fma-define 47.1%

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

      5. associate-/r* 47.1%

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

      6. associate-*r/ 47.1%

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

      7. associate-/r* 47.1%

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

   8. Simplified 47.1%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in ew around inf 69.1%

       \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.9 \cdot 10^{-14} \lor \neg \left(eh \leq 4.8 \cdot 10^{+116}\right):\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.3 \cdot 10^{-104} \lor \neg \left(eh \leq 1.2 \cdot 10^{-84}\right):\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.3e-104) (not (<= eh 1.2e-84))) (fabs eh) (fabs (* ew t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.3e-104) || !(eh <= 1.2e-84)) {
		tmp = fabs(eh);
	} else {
		tmp = fabs((ew * t));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-2.3d-104)) .or. (.not. (eh <= 1.2d-84))) then
        tmp = abs(eh)
    else
        tmp = abs((ew * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.3e-104) || !(eh <= 1.2e-84)) {
		tmp = Math.abs(eh);
	} else {
		tmp = Math.abs((ew * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -2.3e-104) or not (eh <= 1.2e-84):
		tmp = math.fabs(eh)
	else:
		tmp = math.fabs((ew * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.3e-104) || !(eh <= 1.2e-84))
		tmp = abs(eh);
	else
		tmp = abs(Float64(ew * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -2.3e-104) || ~((eh <= 1.2e-84)))
		tmp = abs(eh);
	else
		tmp = abs((ew * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.3e-104], N[Not[LessEqual[eh, 1.2e-84]], $MachinePrecision]], N[Abs[eh], $MachinePrecision], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.2999999999999999e-104 or 1.20000000000000009e-84 < eh

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 51.0%

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

      1. sin-atan 12.3%

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

      2. hypot-1-def 27.6%

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

      3. clear-num 27.6%

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

   7. Applied egg-rr 27.6%

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

   8. Taylor expanded in eh around inf 51.3%

      \[\leadsto \left|\color{blue}{eh}\right| \]

   if -2.2999999999999999e-104 < eh < 1.20000000000000009e-84

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 44.7%

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

      2. fabs-sqr 44.7%

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

      3. add-sqr-sqrt 46.4%

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

      4. expm1-log1p-u 44.3%

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

      5. expm1-undefine 23.3%

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

   6. Applied egg-rr 23.3%

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

      1. expm1-define 44.3%

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

      2. fma-define 44.3%

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

      3. +-commutative 44.3%

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

      4. fma-define 44.3%

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

      5. associate-/r* 44.3%

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

      6. associate-*r/ 44.3%

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

      7. associate-/r* 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in eh around 0 23.3%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + ew \cdot \sin t\right)}\right) \]
   10. Step-by-step derivation

       1. log1p-define 39.3%

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(ew \cdot \sin t\right)}\right) \]

   11. Simplified 39.3%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(ew \cdot \sin t\right)}\right) \]

   12. Taylor expanded in t around 0 24.4%

       \[\leadsto \color{blue}{ew \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative 24.4%

          \[\leadsto \color{blue}{t \cdot ew} \]

   14. Simplified 24.4%

       \[\leadsto \color{blue}{t \cdot ew} \]
   15. Step-by-step derivation

       1. add-sqr-sqrt 23.3%

          \[\leadsto \color{blue}{\sqrt{t \cdot ew} \cdot \sqrt{t \cdot ew}} \]

       2. sqrt-unprod 37.0%

          \[\leadsto \color{blue}{\sqrt{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}} \]

       3. pow2 37.0%

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

   16. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{\sqrt{{\left(t \cdot ew\right)}^{2}}} \]
   17. Step-by-step derivation

       1. unpow2 37.0%

          \[\leadsto \sqrt{\color{blue}{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}} \]

       2. rem-sqrt-square 45.4%

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

   18. Simplified 45.4%

       \[\leadsto \color{blue}{\left|t \cdot ew\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.3 \cdot 10^{-104} \lor \neg \left(eh \leq 1.2 \cdot 10^{-84}\right):\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \left|eh\right| \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(eh)
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[eh], $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. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 39.2%

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

   1. sin-atan 10.3%

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

   2. hypot-1-def 20.4%

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

   3. clear-num 20.4%

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

7. Applied egg-rr 20.4%

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

8. Taylor expanded in eh around inf 39.7%

   \[\leadsto \left|\color{blue}{eh}\right| \]
9. Add Preprocessing

Alternative 10: 10.0% accurate, 306.3× speedup

Math

\[\begin{array}{l} \\ ew \cdot t \end{array} \]

FPCore

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

C

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

Java

public static double code(double eh, double ew, double t) {
	return ew * t;
}

Python

def code(eh, ew, t):
	return ew * t

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(ew * t), $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. fma-define 99.8%

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

   2. associate-/l/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/l/ 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 48.1%

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

   2. fabs-sqr 48.1%

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

   3. add-sqr-sqrt 49.0%

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

   4. expm1-log1p-u 45.9%

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

   5. expm1-undefine 33.1%

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

6. Applied egg-rr 33.1%

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

   1. expm1-define 45.9%

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

   2. fma-define 45.9%

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

   3. +-commutative 45.9%

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

   4. fma-define 45.9%

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

   5. associate-/r* 45.9%

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

   6. associate-*r/ 45.9%

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

   7. associate-/r* 45.9%

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

8. Simplified 45.9%

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

9. Taylor expanded in eh around 0 14.5%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + ew \cdot \sin t\right)}\right) \]
10. Step-by-step derivation

    1. log1p-define 21.1%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(ew \cdot \sin t\right)}\right) \]

11. Simplified 21.1%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(ew \cdot \sin t\right)}\right) \]

12. Taylor expanded in t around 0 12.3%

    \[\leadsto \color{blue}{ew \cdot t} \]
13. Step-by-step derivation

    1. *-commutative 12.3%

       \[\leadsto \color{blue}{t \cdot ew} \]

14. Simplified 12.3%

    \[\leadsto \color{blue}{t \cdot ew} \]

15. Final simplification 12.3%

    \[\leadsto ew \cdot t \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))))))