Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / -ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t}{eh}, \sin t\right)\right|\\ t_2 := \frac{eh \cdot \tan t}{ew}\\ \mathbf{if}\;eh \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.026:\\ \;\;\;\;\left|\left(ew \cdot \cos t + \left(eh \cdot \sin t\right) \cdot t\_2\right) \cdot \frac{-1}{\sqrt{{t\_2}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (fma ew (/ (cos t) eh) (sin t)))))
        (t_2 (/ (* eh (tan t)) ew)))
   (if (<= eh -2.15e-10)
     t_1
     (if (<= eh 0.026)
       (fabs
        (*
         (+ (* ew (cos t)) (* (* eh (sin t)) t_2))
         (/ -1.0 (sqrt (+ (pow t_2 2.0) 1.0)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * fma(ew, (cos(t) / eh), sin(t))));
	double t_2 = (eh * tan(t)) / ew;
	double tmp;
	if (eh <= -2.15e-10) {
		tmp = t_1;
	} else if (eh <= 0.026) {
		tmp = fabs((((ew * cos(t)) + ((eh * sin(t)) * t_2)) * (-1.0 / sqrt((pow(t_2, 2.0) + 1.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * fma(ew, Float64(cos(t) / eh), sin(t))))
	t_2 = Float64(Float64(eh * tan(t)) / ew)
	tmp = 0.0
	if (eh <= -2.15e-10)
		tmp = t_1;
	elseif (eh <= 0.026)
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) + Float64(Float64(eh * sin(t)) * t_2)) * Float64(-1.0 / sqrt(Float64((t_2 ^ 2.0) + 1.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[(ew * N[(N[Cos[t], $MachinePrecision] / eh), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[eh, -2.15e-10], t$95$1, If[LessEqual[eh, 0.026], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[(N[Power[t$95$2, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.15000000000000007e-10 or 0.0259999999999999988 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 66.5%

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

   6. Applied rewrites 66.5%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 66.2%

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

   10. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower-sin.f64 98.9

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

   12. Applied rewrites 98.9%

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

   if -2.15000000000000007e-10 < eh < 0.0259999999999999988

   1. Initial program 99.8%

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

   4. Applied rewrites 98.4%

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

   6. Applied rewrites 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t}{eh}, \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 0.026:\\ \;\;\;\;\left|\left(ew \cdot \cos t + \left(eh \cdot \sin t\right) \cdot \frac{eh \cdot \tan t}{ew}\right) \cdot \frac{-1}{\sqrt{{\left(\frac{eh \cdot \tan t}{ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t}{eh}, \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t}{eh}, \sin t\right)\right|\\ t_2 := \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -900000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.026:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + eh \cdot \left(\sin t \cdot \left(eh \cdot t\_2\right)\right)}{\sqrt{{\left(\left(-eh\right) \cdot t\_2\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (fma ew (/ (cos t) eh) (sin t)))))
        (t_2 (/ (tan t) ew)))
   (if (<= eh -900000000000.0)
     t_1
     (if (<= eh 0.026)
       (fabs
        (/
         (+ (* ew (cos t)) (* eh (* (sin t) (* eh t_2))))
         (sqrt (+ (pow (* (- eh) t_2) 2.0) 1.0))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * fma(ew, (cos(t) / eh), sin(t))));
	double t_2 = tan(t) / ew;
	double tmp;
	if (eh <= -900000000000.0) {
		tmp = t_1;
	} else if (eh <= 0.026) {
		tmp = fabs((((ew * cos(t)) + (eh * (sin(t) * (eh * t_2)))) / sqrt((pow((-eh * t_2), 2.0) + 1.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * fma(ew, Float64(cos(t) / eh), sin(t))))
	t_2 = Float64(tan(t) / ew)
	tmp = 0.0
	if (eh <= -900000000000.0)
		tmp = t_1;
	elseif (eh <= 0.026)
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) + Float64(eh * Float64(sin(t) * Float64(eh * t_2)))) / sqrt(Float64((Float64(Float64(-eh) * t_2) ^ 2.0) + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[(ew * N[(N[Cos[t], $MachinePrecision] / eh), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[eh, -900000000000.0], t$95$1, If[LessEqual[eh, 0.026], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(eh * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[N[((-eh) * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -9e11 or 0.0259999999999999988 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 65.6%

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

   6. Applied rewrites 65.6%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 65.4%

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

   10. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower-sin.f64 98.9

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

   12. Applied rewrites 98.9%

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

   if -9e11 < eh < 0.0259999999999999988

   1. Initial program 99.8%

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

   4. Applied rewrites 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -900000000000:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t}{eh}, \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 0.026:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + eh \cdot \left(\sin t \cdot \left(eh \cdot \frac{\tan t}{ew}\right)\right)}{\sqrt{{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t}{eh}, \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -1.95 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.8 \cdot 10^{+93}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t}{eh}, \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -1.95e+35)
     t_1
     (if (<= ew 1.8e+93) (fabs (* eh (fma ew (/ (cos t) eh) (sin t)))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -1.95e+35) {
		tmp = t_1;
	} else if (ew <= 1.8e+93) {
		tmp = fabs((eh * fma(ew, (cos(t) / eh), sin(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -1.95e+35)
		tmp = t_1;
	elseif (ew <= 1.8e+93)
		tmp = abs(Float64(eh * fma(ew, Float64(cos(t) / eh), sin(t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.95e+35], t$95$1, If[LessEqual[ew, 1.8e+93], N[Abs[N[(eh * N[(ew * N[(N[Cos[t], $MachinePrecision] / eh), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.95e35 or 1.8e93 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 90.1

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

   7. Applied rewrites 90.1%

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

   if -1.95e35 < ew < 1.8e93

   1. Initial program 99.8%

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

   4. Applied rewrites 72.6%

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

   6. Applied rewrites 72.5%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 71.6%

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

   10. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower-sin.f64 98.2

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

   12. Applied rewrites 98.2%

       \[\leadsto \left|\color{blue}{eh \cdot \mathsf{fma}\left(ew, \frac{\cos t}{eh}, \sin t\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -2800000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+137}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -2800000000000.0)
     t_1
     (if (<= eh 4.8e+137) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -2800000000000.0) {
		tmp = t_1;
	} else if (eh <= 4.8e+137) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = 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 = abs((eh * sin(t)))
    if (eh <= (-2800000000000.0d0)) then
        tmp = t_1
    else if (eh <= 4.8d+137) then
        tmp = abs((ew * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (eh <= -2800000000000.0) {
		tmp = t_1;
	} else if (eh <= 4.8e+137) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -2800000000000.0:
		tmp = t_1
	elif eh <= 4.8e+137:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -2800000000000.0)
		tmp = t_1;
	elseif (eh <= 4.8e+137)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -2800000000000.0)
		tmp = t_1;
	elseif (eh <= 4.8e+137)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2800000000000.0], t$95$1, If[LessEqual[eh, 4.8e+137], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.8e12 or 4.79999999999999966e137 < eh

   1. Initial program 99.9%

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

   4. Applied rewrites 59.3%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if -2.8e12 < eh < 4.79999999999999966e137

   1. Initial program 99.8%

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 82.2

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

   7. Applied rewrites 82.2%

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

Alternative 6: 61.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-15}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -2.45e-14)
     t_1
     (if (<= t 2.5e-15) (fabs (fma (* t (* ew -0.5)) t ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -2.45e-14) {
		tmp = t_1;
	} else if (t <= 2.5e-15) {
		tmp = fabs(fma((t * (ew * -0.5)), t, ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (t <= -2.45e-14)
		tmp = t_1;
	elseif (t <= 2.5e-15)
		tmp = abs(fma(Float64(t * Float64(ew * -0.5)), t, ew));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.45e-14], t$95$1, If[LessEqual[t, 2.5e-15], N[Abs[N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.44999999999999997e-14 or 2.5e-15 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 78.2%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 51.9

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

   7. Applied rewrites 51.9%

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

   if -2.44999999999999997e-14 < t < 2.5e-15

   1. Initial program 100.0%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 69.4

          \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, 0.5 \cdot \frac{\color{blue}{eh \cdot eh}}{ew}\right), ew\right)\right| \]

   7. Applied rewrites 69.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, 0.5 \cdot \frac{eh \cdot eh}{ew}\right), ew\right)}\right| \]

   8. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.0

         \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right)\right| \]

   10. Applied rewrites 79.0%

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right)\right| \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 79.0

          \[\leadsto \left|\mathsf{fma}\left(\color{blue}{t \cdot \left(ew \cdot -0.5\right)}, t, ew\right)\right| \]

   12. Applied rewrites 79.0%

       \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 38.7% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (fma (* t (* ew -0.5)) t ew)))

C

double code(double eh, double ew, double t) {
	return fabs(fma((t * (ew * -0.5)), t, ew));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(t * Float64(ew * -0.5)), t, ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 83.7%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 37.5

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, 0.5 \cdot \frac{\color{blue}{eh \cdot eh}}{ew}\right), ew\right)\right| \]

7. Applied rewrites 37.5%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, 0.5 \cdot \frac{eh \cdot eh}{ew}\right), ew\right)}\right| \]

8. Taylor expanded in ew around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 42.5

      \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right)\right| \]

10. Applied rewrites 42.5%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right)\right| \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-*.f64 42.5

       \[\leadsto \left|\mathsf{fma}\left(\color{blue}{t \cdot \left(ew \cdot -0.5\right)}, t, ew\right)\right| \]

12. Applied rewrites 42.5%

    \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)}\right| \]
13. Add Preprocessing

Alternative 8: 38.7% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(t \cdot t, ew \cdot -0.5, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (fma (* t t) (* ew -0.5) ew)))

C

double code(double eh, double ew, double t) {
	return fabs(fma((t * t), (ew * -0.5), ew));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(t * t), Float64(ew * -0.5), ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(t * t), $MachinePrecision] * N[(ew * -0.5), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 83.7%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 37.5

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, 0.5 \cdot \frac{\color{blue}{eh \cdot eh}}{ew}\right), ew\right)\right| \]

7. Applied rewrites 37.5%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, 0.5 \cdot \frac{eh \cdot eh}{ew}\right), ew\right)}\right| \]

8. Taylor expanded in ew around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 42.5

      \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right)\right| \]

10. Applied rewrites 42.5%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right)\right| \]
11. Add Preprocessing

Alternative 9: 4.9% accurate, 47.9× speedup

Math

\[\begin{array}{l} \\ \left|-0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* -0.5 (* ew (* t t)))))

C

double code(double eh, double ew, double t) {
	return fabs((-0.5 * (ew * (t * 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(((-0.5d0) * (ew * (t * t))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((-0.5 * (ew * (t * t))));
}

Python

def code(eh, ew, t):
	return math.fabs((-0.5 * (ew * (t * t))))

Julia

function code(eh, ew, t)
	return abs(Float64(-0.5 * Float64(ew * Float64(t * t))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((-0.5 * (ew * (t * t))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 83.7%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 37.5

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, 0.5 \cdot \frac{\color{blue}{eh \cdot eh}}{ew}\right), ew\right)\right| \]

7. Applied rewrites 37.5%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, 0.5 \cdot \frac{eh \cdot eh}{ew}\right), ew\right)}\right| \]

8. Taylor expanded in ew around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 42.5

      \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right)\right| \]

10. Applied rewrites 42.5%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \color{blue}{ew \cdot -0.5}, ew\right)\right| \]

11. Taylor expanded in t around inf

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 4.7

       \[\leadsto \left|-0.5 \cdot \left(ew \cdot \color{blue}{\left(t \cdot t\right)}\right)\right| \]

13. Applied rewrites 4.7%

    \[\leadsto \left|\color{blue}{-0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))