Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.5s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / tan(t);
	tmp = abs((((eh * cos(t)) * sin(atan(t_1))) + ((ew * sin(t)) / hypot(1.0, 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[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $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. *-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \frac{ew \cdot \sin t}{\mathsf{hypot}\left(1, \frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]
8. 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. *-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in ew around inf 98.3%

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

Alternative 4: 85.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -6.6 \cdot 10^{-78} \lor \neg \left(ew \leq 3.1 \cdot 10^{+25}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -6.6e-78) (not (<= ew 3.1e+25)))
   (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 <= -6.6e-78) || !(ew <= 3.1e+25)) {
		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 <= (-6.6d-78)) .or. (.not. (ew <= 3.1d+25))) 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 <= -6.6e-78) || !(ew <= 3.1e+25)) {
		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 <= -6.6e-78) or not (ew <= 3.1e+25):
		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 <= -6.6e-78) || !(ew <= 3.1e+25))
		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 <= -6.6e-78) || ~((ew <= 3.1e+25)))
		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, -6.6e-78], N[Not[LessEqual[ew, 3.1e+25]], $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 < -6.59999999999999963e-78 or 3.0999999999999998e25 < 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. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-/r* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 87.3%

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

   8. Taylor expanded in ew around inf 86.7%

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

   if -6.59999999999999963e-78 < ew < 3.0999999999999998e25

   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 87.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.6 \cdot 10^{-78} \lor \neg \left(ew \leq 3.1 \cdot 10^{+25}\right):\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -3 \cdot 10^{+38} \lor \neg \left(ew \leq 1.05 \cdot 10^{+190}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -3e+38) (not (<= ew 1.05e+190)))
   (fabs (* ew (sin t)))
   (fabs (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3e+38) || !(ew <= 1.05e+190)) {
		tmp = fabs((ew * sin(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 <= (-3d+38)) .or. (.not. (ew <= 1.05d+190))) then
        tmp = abs((ew * sin(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 <= -3e+38) || !(ew <= 1.05e+190)) {
		tmp = Math.abs((ew * Math.sin(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 <= -3e+38) or not (ew <= 1.05e+190):
		tmp = math.fabs((ew * math.sin(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 <= -3e+38) || !(ew <= 1.05e+190))
		tmp = abs(Float64(ew * sin(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 <= -3e+38) || ~((ew <= 1.05e+190)))
		tmp = abs((ew * sin(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, -3e+38], N[Not[LessEqual[ew, 1.05e+190]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $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 < -3.0000000000000001e38 or 1.05e190 < ew

   1. Initial program 99.8%

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

      1. 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-cube-cbrt 97.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|} \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right) \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}} \]

      2. pow3 97.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right)}^{3}} \]

   6. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, \cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 26.9%

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

      1. *-commutative 26.9%

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

   9. Simplified 26.9%

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

       1. add-sqr-sqrt 26.3%

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

       2. sqrt-unprod 38.3%

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

       3. pow2 38.3%

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

       4. *-commutative 38.3%

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

   11. Applied egg-rr 38.3%

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

       1. unpow2 38.3%

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

       2. rem-sqrt-square 75.8%

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

   13. Simplified 75.8%

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

   if -3.0000000000000001e38 < ew < 1.05e190

   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 79.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3 \cdot 10^{+38} \lor \neg \left(ew \leq 1.05 \cdot 10^{+190}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-18} \lor \neg \left(t \leq 1.9 \cdot 10^{-55}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -1.2e-18) (not (<= t 1.9e-55)))
   (fabs (* ew (sin t)))
   (fabs (* eh (sin (atan (/ eh (* ew t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.2e-18) || !(t <= 1.9e-55)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((eh * sin(atan((eh / (ew * t))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.2d-18)) .or. (.not. (t <= 1.9d-55))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs((eh * sin(atan((eh / (ew * t))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.2e-18) || !(t <= 1.9e-55)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -1.2e-18) or not (t <= 1.9e-55):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((eh * math.sin(math.atan((eh / (ew * t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -1.2e-18) || !(t <= 1.9e-55))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(eh * sin(atan(Float64(eh / Float64(ew * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -1.2e-18) || ~((t <= 1.9e-55)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((eh * sin(atan((eh / (ew * t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -1.2e-18], N[Not[LessEqual[t, 1.9e-55]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.19999999999999997e-18 or 1.8999999999999998e-55 < t

   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. Step-by-step derivation

      1. add-cube-cbrt 97.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|} \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right) \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}} \]

      2. pow3 97.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right)}^{3}} \]

   6. Applied egg-rr 45.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, \cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 19.4%

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

      1. *-commutative 19.4%

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

   9. Simplified 19.4%

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

       1. add-sqr-sqrt 18.5%

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

       2. sqrt-unprod 25.3%

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

       3. pow2 25.3%

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

       4. *-commutative 25.3%

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

   11. Applied egg-rr 25.3%

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

       1. unpow2 25.3%

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

       2. rem-sqrt-square 45.9%

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

   13. Simplified 45.9%

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

   if -1.19999999999999997e-18 < t < 1.8999999999999998e-55

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 76.2%

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

   6. Taylor expanded in t around 0 76.2%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-18} \lor \neg \left(t \leq 1.9 \cdot 10^{-55}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-156} \lor \neg \left(t \leq 4 \cdot 10^{-77}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -4.2e-156) (not (<= t 4e-77)))
   (fabs (* ew (sin t)))
   (* eh (sin (atan (/ (/ eh t) ew))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -4.2e-156) || !(t <= 4e-77)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = eh * sin(atan(((eh / t) / ew)));
	}
	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 ((t <= (-4.2d-156)) .or. (.not. (t <= 4d-77))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = eh * sin(atan(((eh / t) / ew)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -4.2e-156) || !(t <= 4e-77)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = eh * Math.sin(Math.atan(((eh / t) / ew)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -4.2e-156) or not (t <= 4e-77):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = eh * math.sin(math.atan(((eh / t) / ew)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -4.2e-156) || !(t <= 4e-77))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = Float64(eh * sin(atan(Float64(Float64(eh / t) / ew))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -4.2e-156) || ~((t <= 4e-77)))
		tmp = abs((ew * sin(t)));
	else
		tmp = eh * sin(atan(((eh / t) / ew)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -4.2e-156], N[Not[LessEqual[t, 4e-77]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(eh * N[Sin[N[ArcTan[N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.20000000000000025e-156 or 3.9999999999999997e-77 < t

   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. Step-by-step derivation

      1. add-cube-cbrt 97.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|} \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right) \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}} \]

      2. pow3 97.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right)}^{3}} \]

   6. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, \cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 18.5%

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

      1. *-commutative 18.5%

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

   9. Simplified 18.5%

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

       1. add-sqr-sqrt 17.5%

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

       2. sqrt-unprod 25.6%

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

       3. pow2 25.6%

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

       4. *-commutative 25.6%

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

   11. Applied egg-rr 25.6%

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

       1. unpow2 25.6%

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

       2. rem-sqrt-square 44.6%

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

   13. Simplified 44.6%

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

   if -4.20000000000000025e-156 < t < 3.9999999999999997e-77

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

      2. associate-/l/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 82.0%

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

   6. Taylor expanded in t around 0 82.0%

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

      1. add-sqr-sqrt 81.5%

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

      2. pow2 81.5%

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

      3. *-commutative 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. add-sqr-sqrt 43.7%

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

      2. fabs-sqr 43.7%

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

      3. add-sqr-sqrt 43.7%

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

      4. pow2 43.7%

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

      5. add-sqr-sqrt 44.8%

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

      6. *-commutative 44.8%

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

      7. associate-/r* 44.8%

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

   10. Applied egg-rr 44.8%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-156} \lor \neg \left(t \leq 4 \cdot 10^{-77}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{t}}{ew}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \sin t\right| \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $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. 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-cube-cbrt 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|} \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right) \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}} \]

   2. pow3 97.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right)}^{3}} \]

6. Applied egg-rr 44.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, \cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right)}^{3}} \]

7. Taylor expanded in ew around inf 16.3%

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

   1. *-commutative 16.3%

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

9. Simplified 16.3%

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

    1. add-sqr-sqrt 15.4%

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

    2. sqrt-unprod 23.6%

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

    3. pow2 23.6%

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

    4. *-commutative 23.6%

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

11. Applied egg-rr 23.6%

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

    1. unpow2 23.6%

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

    2. rem-sqrt-square 38.6%

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

13. Simplified 38.6%

    \[\leadsto \color{blue}{\left|ew \cdot \sin t\right|} \]
14. Add Preprocessing

Alternative 9: 21.6% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(ew * N[Sin[t], $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. 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-cube-cbrt 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|} \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right) \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}} \]

   2. pow3 97.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right)}^{3}} \]

6. Applied egg-rr 44.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, \cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right)}^{3}} \]

7. Taylor expanded in ew around inf 16.3%

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

   1. *-commutative 16.3%

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

9. Simplified 16.3%

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

10. Final simplification 16.3%

    \[\leadsto ew \cdot \sin t \]
11. Add Preprocessing

Alternative 10: 10.3% 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-cube-cbrt 97.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|} \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right) \cdot \sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}} \]

   2. pow3 97.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\mathsf{fma}\left(ew \cdot \sin t, \cos \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right), eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)\right|}\right)}^{3}} \]

6. Applied egg-rr 44.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\sin t}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, \cos t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right)}^{3}} \]

7. Taylor expanded in ew around inf 16.3%

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

   1. *-commutative 16.3%

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

9. Simplified 16.3%

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

10. Taylor expanded in t around 0 8.1%

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

    1. *-commutative 8.1%

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

12. Simplified 8.1%

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

13. Final simplification 8.1%

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

Reproduce

herbie shell --seed 2024113 
(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))))))))