Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.1s

Alternatives: 10

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: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \left|eh \cdot \mathsf{fma}\left(\cos t, \frac{ew}{eh}, \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{if}\;eh \leq -2 \cdot 10^{-41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 0.46:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + eh \cdot \left(\sin t \cdot \left(eh \cdot t\_1\right)\right)}{\sqrt{1 + {\left(\left(-eh\right) \cdot t\_1\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2
         (fabs
          (*
           eh
           (fma
            (cos t)
            (/ ew eh)
            (* (sin (atan (/ (* eh (tan t)) (- ew)))) (- (sin t))))))))
   (if (<= eh -2e-41)
     t_2
     (if (<= eh 0.46)
       (fabs
        (/
         (+ (* ew (cos t)) (* eh (* (sin t) (* eh t_1))))
         (sqrt (+ 1.0 (pow (* (- eh) t_1) 2.0)))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = fabs((eh * fma(cos(t), (ew / eh), (sin(atan(((eh * tan(t)) / -ew))) * -sin(t)))));
	double tmp;
	if (eh <= -2e-41) {
		tmp = t_2;
	} else if (eh <= 0.46) {
		tmp = fabs((((ew * cos(t)) + (eh * (sin(t) * (eh * t_1)))) / sqrt((1.0 + pow((-eh * t_1), 2.0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = abs(Float64(eh * fma(cos(t), Float64(ew / eh), Float64(sin(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))) * Float64(-sin(t))))))
	tmp = 0.0
	if (eh <= -2e-41)
		tmp = t_2;
	elseif (eh <= 0.46)
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) + Float64(eh * Float64(sin(t) * Float64(eh * t_1)))) / sqrt(Float64(1.0 + (Float64(Float64(-eh) * t_1) ^ 2.0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[(ew / eh), $MachinePrecision] + N[(N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2e-41], t$95$2, If[LessEqual[eh, 0.46], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(eh * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[((-eh) * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.00000000000000001e-41 or 0.46000000000000002 < 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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 25.0

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

   5. Applied rewrites 25.0%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 99.7%

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

   if -2.00000000000000001e-41 < eh < 0.46000000000000002

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{-41}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(\cos t, \frac{ew}{eh}, \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{elif}\;eh \leq 0.46:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + eh \cdot \left(\sin t \cdot \left(eh \cdot \frac{\tan t}{ew}\right)\right)}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(\cos t, \frac{ew}{eh}, \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -eh \cdot \frac{t}{ew}\\ t_2 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -8 \cdot 10^{+82}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 4.2 \cdot 10^{-46}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;eh \leq 4.9 \cdot 10^{+134}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(\frac{ew}{eh \cdot \sqrt{1 + {\left(\frac{eh \cdot \tan t}{ew}\right)}^{2}}}, \cos t, -\frac{\sin t \cdot t\_1}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (* eh (/ t ew)))) (t_2 (fabs (* eh (sin t)))))
   (if (<= eh -8e+82)
     t_2
     (if (<= eh 4.2e-46)
       (fabs (* ew (cos t)))
       (if (<= eh 4.9e+134)
         (fabs
          (*
           eh
           (fma
            (/ ew (* eh (sqrt (+ 1.0 (pow (/ (* eh (tan t)) ew) 2.0)))))
            (cos t)
            (- (/ (* (sin t) t_1) (sqrt (fma t_1 t_1 1.0)))))))
         t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = -(eh * (t / ew));
	double t_2 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -8e+82) {
		tmp = t_2;
	} else if (eh <= 4.2e-46) {
		tmp = fabs((ew * cos(t)));
	} else if (eh <= 4.9e+134) {
		tmp = fabs((eh * fma((ew / (eh * sqrt((1.0 + pow(((eh * tan(t)) / ew), 2.0))))), cos(t), -((sin(t) * t_1) / sqrt(fma(t_1, t_1, 1.0))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(-Float64(eh * Float64(t / ew)))
	t_2 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -8e+82)
		tmp = t_2;
	elseif (eh <= 4.2e-46)
		tmp = abs(Float64(ew * cos(t)));
	elseif (eh <= 4.9e+134)
		tmp = abs(Float64(eh * fma(Float64(ew / Float64(eh * sqrt(Float64(1.0 + (Float64(Float64(eh * tan(t)) / ew) ^ 2.0))))), cos(t), Float64(-Float64(Float64(sin(t) * t_1) / sqrt(fma(t_1, t_1, 1.0)))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[(eh * N[(t / ew), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -8e+82], t$95$2, If[LessEqual[eh, 4.2e-46], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.9e+134], N[Abs[N[(eh * N[(N[(ew / N[(eh * N[Sqrt[N[(1.0 + N[Power[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision] + (-N[(N[(N[Sin[t], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -7.9999999999999997e82 or 4.89999999999999996e134 < 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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if -7.9999999999999997e82 < eh < 4.19999999999999975e-46

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 84.2

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

   7. Applied rewrites 84.2%

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

   if 4.19999999999999975e-46 < eh < 4.89999999999999996e134

   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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 99.5%

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

   13. Applied rewrites 77.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8 \cdot 10^{+82}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;eh \leq 4.2 \cdot 10^{-46}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;eh \leq 4.9 \cdot 10^{+134}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(\frac{ew}{eh \cdot \sqrt{1 + {\left(\frac{eh \cdot \tan t}{ew}\right)}^{2}}}, \cos t, -\frac{\sin t \cdot \left(-eh \cdot \frac{t}{ew}\right)}{\sqrt{\mathsf{fma}\left(-eh \cdot \frac{t}{ew}, -eh \cdot \frac{t}{ew}, 1\right)}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \mathsf{fma}\left(\cos t, \frac{ew}{eh}, \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{if}\;eh \leq -2.1 \cdot 10^{-185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.8 \cdot 10^{-46}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           eh
           (fma
            (cos t)
            (/ ew eh)
            (* (sin (atan (/ (* eh (tan t)) (- ew)))) (- (sin t))))))))
   (if (<= eh -2.1e-185) t_1 (if (<= eh 3.8e-46) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * fma(cos(t), (ew / eh), (sin(atan(((eh * tan(t)) / -ew))) * -sin(t)))));
	double tmp;
	if (eh <= -2.1e-185) {
		tmp = t_1;
	} else if (eh <= 3.8e-46) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * fma(cos(t), Float64(ew / eh), Float64(sin(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))) * Float64(-sin(t))))))
	tmp = 0.0
	if (eh <= -2.1e-185)
		tmp = t_1;
	elseif (eh <= 3.8e-46)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[(ew / eh), $MachinePrecision] + N[(N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.1e-185], t$95$1, If[LessEqual[eh, 3.8e-46], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.1e-185 or 3.7999999999999997e-46 < 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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 97.6%

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

   10. Applied rewrites 97.6%

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

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 97.1%

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

   if -2.1e-185 < eh < 3.7999999999999997e-46

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 94.1

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

   7. Applied rewrites 94.1%

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

Alternative 5: 74.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -1.25 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -1.25e+23) t_1 (if (<= ew 4.6e-11) (fabs (* 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 <= -1.25e+23) {
		tmp = t_1;
	} else if (ew <= 4.6e-11) {
		tmp = fabs((eh * sin(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((ew * cos(t)))
    if (ew <= (-1.25d+23)) then
        tmp = t_1
    else if (ew <= 4.6d-11) then
        tmp = abs((eh * sin(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((ew * Math.cos(t)));
	double tmp;
	if (ew <= -1.25e+23) {
		tmp = t_1;
	} else if (ew <= 4.6e-11) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -1.25e+23:
		tmp = t_1
	elif ew <= 4.6e-11:
		tmp = math.fabs((eh * math.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 <= -1.25e+23)
		tmp = t_1;
	elseif (ew <= 4.6e-11)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -1.25e+23)
		tmp = t_1;
	elseif (ew <= 4.6e-11)
		tmp = abs((eh * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.25e+23], t$95$1, If[LessEqual[ew, 4.6e-11], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.25e23 or 4.60000000000000027e-11 < 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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -1.25e23 < ew < 4.60000000000000027e-11

   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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 70.4

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

   7. Applied rewrites 70.4%

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

Alternative 6: 58.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -3.8 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.00029:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot \left(0.5 \cdot \frac{eh \cdot eh}{ew}\right), t, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -3.8e+14)
     t_1
     (if (<= eh 0.00029)
       (fabs (fma (* t (* 0.5 (/ (* eh eh) ew))) t ew))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -3.8e+14) {
		tmp = t_1;
	} else if (eh <= 0.00029) {
		tmp = fabs(fma((t * (0.5 * ((eh * eh) / ew))), 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 (eh <= -3.8e+14)
		tmp = t_1;
	elseif (eh <= 0.00029)
		tmp = abs(fma(Float64(t * Float64(0.5 * Float64(Float64(eh * eh) / ew))), 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[eh, -3.8e+14], t$95$1, If[LessEqual[eh, 0.00029], N[Abs[N[(N[(t * N[(0.5 * N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.8e14 or 2.9e-4 < 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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 68.6

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

   7. Applied rewrites 68.6%

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

   if -3.8e14 < eh < 2.9e-4

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 46.5%

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

   9. Applied rewrites 46.5%

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

   10. Taylor expanded in eh around inf

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

   12. Applied rewrites 52.7%

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

Alternative 7: 40.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{0.5 \cdot \left(t \cdot \left(eh \cdot eh\right)\right)}{ew}, t, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -1.35)
   (fabs (fma (/ (* 0.5 (* t (* eh eh))) ew) t ew))
   (fabs (fma (* t (* ew -0.5)) t ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -1.35) {
		tmp = fabs(fma(((0.5 * (t * (eh * eh))) / ew), t, ew));
	} else {
		tmp = fabs(fma((t * (ew * -0.5)), t, ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -1.35)
		tmp = abs(fma(Float64(Float64(0.5 * Float64(t * Float64(eh * eh))) / ew), t, ew));
	else
		tmp = abs(fma(Float64(t * Float64(ew * -0.5)), t, ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -1.35], N[Abs[N[(N[(N[(0.5 * N[(t * N[(eh * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3500000000000001

   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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 5.5%

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

   9. Applied rewrites 5.6%

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

   10. Taylor expanded in eh around inf

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

   12. Applied rewrites 12.3%

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

   if -1.3500000000000001 < t

   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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 39.2%

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

   9. Applied rewrites 39.7%

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

   10. Taylor expanded in eh around 0

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

   12. Applied rewrites 46.3%

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

Alternative 8: 39.2% 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
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 76.0%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 N/A

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

7. Applied rewrites 29.6%

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

9. Applied rewrites 30.0%

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

10. Taylor expanded in eh around 0

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

12. Applied rewrites 34.7%

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

Alternative 9: 39.1% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \mathsf{fma}\left(-0.5, t \cdot t, 1\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* ew (fma -0.5 (* t t) 1.0))))

C

double code(double eh, double ew, double t) {
	return fabs((ew * fma(-0.5, (t * t), 1.0)));
}

Julia

function code(eh, ew, t)
	return abs(Float64(ew * fma(-0.5, Float64(t * t), 1.0)))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 76.0%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 N/A

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

7. Applied rewrites 29.6%

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

8. Taylor expanded in ew around inf

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

10. Applied rewrites 34.7%

    \[\leadsto \left|ew \cdot \color{blue}{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}\right| \]
11. Add Preprocessing

Alternative 10: 4.8% 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
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 76.0%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 N/A

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

7. Applied rewrites 29.6%

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

8. Taylor expanded in ew around inf

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

10. Applied rewrites 34.7%

    \[\leadsto \left|ew \cdot \color{blue}{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}\right| \]

11. Taylor expanded in t around inf

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

13. Applied rewrites 5.0%

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

Reproduce

herbie shell --seed 2024219 
(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)))))))