Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.6s

Alternatives: 11

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

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

Alternative 2: 52.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq 10^{-247}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t \cdot eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right) \cdot \left(-ew\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) 1e-247)
     (fabs
      (* (cos (atan (/ (* (sin t) eh) (* (fma (* t t) -0.5 1.0) (- ew))))) ew))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 1e-247) {
		tmp = fabs((cos(atan(((sin(t) * eh) / (fma((t * t), -0.5, 1.0) * -ew)))) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= 1e-247)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) * eh) / Float64(fma(Float64(t * t), -0.5, 1.0) * Float64(-ew))))) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-247], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] / N[(N[(N[(t * t), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 1e-247

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 44.6%

      \[\leadsto \left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right)}\right) \cdot ew\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

       \[\leadsto \left|\cos \tan^{-1} \left(\frac{\sin t \cdot \left(-eh\right)}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right) \cdot ew}\right) \cdot ew\right| \]

   if 1e-247 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.9%

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 61.8

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

   10. Applied rewrites 61.8%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq 10^{-247}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t \cdot eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right) \cdot \left(-ew\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -5e-295)
     (fabs (* (cos (atan (* (/ (tan t) ew) eh))) ew))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -5e-295) {
		tmp = fabs((cos(atan(((tan(t) / ew) * eh))) * ew));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-5d-295)) then
        tmp = abs((cos(atan(((tan(t) / ew) * eh))) * ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -5e-295) {
		tmp = Math.abs((Math.cos(Math.atan(((Math.tan(t) / ew) * eh))) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -5e-295:
		tmp = math.fabs((math.cos(math.atan(((math.tan(t) / ew) * eh))) * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -5e-295)
		tmp = abs(Float64(cos(atan(Float64(Float64(tan(t) / ew) * eh))) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -5e-295)
		tmp = abs((cos(atan(((tan(t) / ew) * eh))) * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-295], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -5.00000000000000008e-295

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 44.0%

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

   if -5.00000000000000008e-295 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.9%

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 28.6

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

   7. Applied rewrites 28.6%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 62.0

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

   10. Applied rewrites 62.0%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-295}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -5 \cdot 10^{-295}:\\ \;\;\;\;\left|\sin \left(\frac{\mathsf{PI}\left(\right)}{2} + \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -5e-295)
     (fabs (* (sin (+ (/ (PI) 2.0) (atan (/ (* eh t) ew)))) ew))
     t_1)))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -5.00000000000000008e-295

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 42.0%

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

   if -5.00000000000000008e-295 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.9%

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 28.6

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

   7. Applied rewrites 28.6%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 62.0

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

   10. Applied rewrites 62.0%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-295}:\\ \;\;\;\;\left|\sin \left(\frac{\mathsf{PI}\left(\right)}{2} + \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-t) * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $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. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 86.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.9 \cdot 10^{-43} \lor \neg \left(eh \leq 1.22 \cdot 10^{-5}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right) - \left(ew \cdot 1\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.9e-43) (not (<= eh 1.22e-5)))
   (fabs
    (-
     (* (* eh (sin t)) (sin (atan (/ (* (- t) eh) ew))))
     (* (* ew 1.0) (cos (atan (/ (* eh (tan t)) (- ew)))))))
   (fabs
    (* (cos (atan (* (/ (- (sin t)) ew) (/ eh (cos t))))) (* (cos t) ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.9e-43) || !(eh <= 1.22e-5)) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-t * eh) / ew)))) - ((ew * 1.0) * cos(atan(((eh * tan(t)) / -ew))))));
	} else {
		tmp = fabs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-2.9d-43)) .or. (.not. (eh <= 1.22d-5))) then
        tmp = abs((((eh * sin(t)) * sin(atan(((-t * eh) / ew)))) - ((ew * 1.0d0) * cos(atan(((eh * tan(t)) / -ew))))))
    else
        tmp = abs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.9e-43) || !(eh <= 1.22e-5)) {
		tmp = Math.abs((((eh * Math.sin(t)) * Math.sin(Math.atan(((-t * eh) / ew)))) - ((ew * 1.0) * Math.cos(Math.atan(((eh * Math.tan(t)) / -ew))))));
	} else {
		tmp = Math.abs((Math.cos(Math.atan(((-Math.sin(t) / ew) * (eh / Math.cos(t))))) * (Math.cos(t) * ew)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -2.9e-43) or not (eh <= 1.22e-5):
		tmp = math.fabs((((eh * math.sin(t)) * math.sin(math.atan(((-t * eh) / ew)))) - ((ew * 1.0) * math.cos(math.atan(((eh * math.tan(t)) / -ew))))))
	else:
		tmp = math.fabs((math.cos(math.atan(((-math.sin(t) / ew) * (eh / math.cos(t))))) * (math.cos(t) * ew)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.9e-43) || !(eh <= 1.22e-5))
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-t) * eh) / ew)))) - Float64(Float64(ew * 1.0) * cos(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))))));
	else
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(-sin(t)) / ew) * Float64(eh / cos(t))))) * Float64(cos(t) * ew)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -2.9e-43) || ~((eh <= 1.22e-5)))
		tmp = abs((((eh * sin(t)) * sin(atan(((-t * eh) / ew)))) - ((ew * 1.0) * cos(atan(((eh * tan(t)) / -ew))))));
	else
		tmp = abs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.9e-43], N[Not[LessEqual[eh, 1.22e-5]], $MachinePrecision]], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-t) * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * 1.0), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.9000000000000001e-43 or 1.22000000000000001e-5 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.1%

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

   if -2.9000000000000001e-43 < eh < 1.22000000000000001e-5

   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 eh around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 93.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.9 \cdot 10^{-43} \lor \neg \left(eh \leq 1.22 \cdot 10^{-5}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right) - \left(ew \cdot 1\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -4.4 \cdot 10^{-5} \lor \neg \left(eh \leq 2 \cdot 10^{+74}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(t \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \left(t \cdot t\right), -eh\right)}{ew}\right) \cdot \sin t\right) \cdot \left(-eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -4.4e-5) (not (<= eh 2e+74)))
   (fabs
    (*
     (*
      (tanh
       (asinh (* t (/ (fma -0.3333333333333333 (* eh (* t t)) (- eh)) ew))))
      (sin t))
     (- eh)))
   (fabs
    (* (cos (atan (* (/ (- (sin t)) ew) (/ eh (cos t))))) (* (cos t) ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -4.4e-5) || !(eh <= 2e+74)) {
		tmp = fabs(((tanh(asinh((t * (fma(-0.3333333333333333, (eh * (t * t)), -eh) / ew)))) * sin(t)) * -eh));
	} else {
		tmp = fabs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -4.4e-5) || !(eh <= 2e+74))
		tmp = abs(Float64(Float64(tanh(asinh(Float64(t * Float64(fma(-0.3333333333333333, Float64(eh * Float64(t * t)), Float64(-eh)) / ew)))) * sin(t)) * Float64(-eh)));
	else
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(-sin(t)) / ew) * Float64(eh / cos(t))))) * Float64(cos(t) * ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -4.4e-5], N[Not[LessEqual[eh, 2e+74]], $MachinePrecision]], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(t * N[(N[(-0.3333333333333333 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] + (-eh)), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.3999999999999999e-5 or 1.9999999999999999e74 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. distribute-lft-neg-in N/A

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\left(\tanh \sinh^{-1} \left(t \cdot \left(-1 \cdot \frac{eh}{ew} + \frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew}\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 71.4%

       \[\leadsto \left|\left(\tanh \sinh^{-1} \left(t \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \left(t \cdot t\right), -1 \cdot eh\right)}{ew}\right) \cdot \left(-\sin t\right)\right) \cdot eh\right| \]

   if -4.3999999999999999e-5 < eh < 1.9999999999999999e74

   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 eh around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 90.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.4 \cdot 10^{-5} \lor \neg \left(eh \leq 2 \cdot 10^{+74}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(t \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \left(t \cdot t\right), -eh\right)}{ew}\right) \cdot \sin t\right) \cdot \left(-eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.55 \cdot 10^{-5} \lor \neg \left(eh \leq 1.85 \cdot 10^{+74}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(t \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \left(t \cdot t\right), -eh\right)}{ew}\right) \cdot \sin t\right) \cdot \left(-eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t \cdot eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right) \cdot \left(-ew\right)}\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.55e-5) (not (<= eh 1.85e+74)))
   (fabs
    (*
     (*
      (tanh
       (asinh (* t (/ (fma -0.3333333333333333 (* eh (* t t)) (- eh)) ew))))
      (sin t))
     (- eh)))
   (fabs
    (* (cos (atan (/ (* (sin t) eh) (* (fma (* t t) -0.5 1.0) (- ew))))) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.55e-5) || !(eh <= 1.85e+74)) {
		tmp = fabs(((tanh(asinh((t * (fma(-0.3333333333333333, (eh * (t * t)), -eh) / ew)))) * sin(t)) * -eh));
	} else {
		tmp = fabs((cos(atan(((sin(t) * eh) / (fma((t * t), -0.5, 1.0) * -ew)))) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.55e-5) || !(eh <= 1.85e+74))
		tmp = abs(Float64(Float64(tanh(asinh(Float64(t * Float64(fma(-0.3333333333333333, Float64(eh * Float64(t * t)), Float64(-eh)) / ew)))) * sin(t)) * Float64(-eh)));
	else
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) * eh) / Float64(fma(Float64(t * t), -0.5, 1.0) * Float64(-ew))))) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.55e-5], N[Not[LessEqual[eh, 1.85e+74]], $MachinePrecision]], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(t * N[(N[(-0.3333333333333333 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] + (-eh)), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] / N[(N[(N[(t * t), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.54999999999999998e-5 or 1.8500000000000001e74 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. distribute-lft-neg-in N/A

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\left(\tanh \sinh^{-1} \left(t \cdot \left(-1 \cdot \frac{eh}{ew} + \frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew}\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 71.4%

       \[\leadsto \left|\left(\tanh \sinh^{-1} \left(t \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \left(t \cdot t\right), -1 \cdot eh\right)}{ew}\right) \cdot \left(-\sin t\right)\right) \cdot eh\right| \]

   if -2.54999999999999998e-5 < eh < 1.8500000000000001e74

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.6%

      \[\leadsto \left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right)}\right) \cdot ew\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 57.6%

       \[\leadsto \left|\cos \tan^{-1} \left(\frac{\sin t \cdot \left(-eh\right)}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right) \cdot ew}\right) \cdot ew\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.55 \cdot 10^{-5} \lor \neg \left(eh \leq 1.85 \cdot 10^{+74}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(t \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \left(t \cdot t\right), -eh\right)}{ew}\right) \cdot \sin t\right) \cdot \left(-eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t \cdot eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right) \cdot \left(-ew\right)}\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.2% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 36.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 14.8

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

7. Applied rewrites 14.8%

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

8. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 30.7

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

10. Applied rewrites 30.7%

    \[\leadsto \color{blue}{ew \cdot \cos t} \]
11. Add Preprocessing

Alternative 10: 20.8% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 36.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 14.8

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

7. Applied rewrites 14.8%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 16.7%

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

12. Applied rewrites 16.8%

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

Alternative 11: 20.7% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 36.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 14.8

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

7. Applied rewrites 14.8%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 16.7%

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

11. Taylor expanded in eh around 0

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

13. Applied rewrites 16.7%

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

Reproduce

herbie shell --seed 2024337 
(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)))))))