Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.6s

Alternatives: 7

Speedup: N/A×

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, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \frac{\tan t}{ew}\\ \left|\mathsf{fma}\left(\frac{\cos t}{\sqrt{1 + {t\_1}^{2}}}, ew, \left(-eh \cdot \sin t\right) \cdot \sin \tan^{-1} t\_1\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (/ (tan t) ew))))
   (fabs
    (fma
     (/ (cos t) (sqrt (+ 1.0 (pow t_1 2.0))))
     ew
     (* (- (* eh (sin t))) (sin (atan t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * (tan(t) / ew);
	return fabs(fma((cos(t) / sqrt((1.0 + pow(t_1, 2.0)))), ew, (-(eh * sin(t)) * sin(atan(t_1)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * Float64(tan(t) / ew))
	return abs(fma(Float64(cos(t) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))), ew, Float64(Float64(-Float64(eh * sin(t))) * sin(atan(t_1)))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{if}\;ew \leq -8.5 \cdot 10^{-191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.8 \cdot 10^{-282}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           ew
           (fma
            eh
            (/ (* (sin t) (sin (atan (* (- eh) (/ (tan t) ew))))) ew)
            (- (cos t)))))))
   (if (<= ew -8.5e-191)
     t_1
     (if (<= ew 3.8e-282)
       (fabs (* (* eh (sin t)) (sin (atan (/ (* eh (tan t)) (- ew))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * fma(eh, ((sin(t) * sin(atan((-eh * (tan(t) / ew))))) / ew), -cos(t))));
	double tmp;
	if (ew <= -8.5e-191) {
		tmp = t_1;
	} else if (ew <= 3.8e-282) {
		tmp = fabs(((eh * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * fma(eh, Float64(Float64(sin(t) * sin(atan(Float64(Float64(-eh) * Float64(tan(t) / ew))))) / ew), Float64(-cos(t)))))
	tmp = 0.0
	if (ew <= -8.5e-191)
		tmp = t_1;
	elseif (ew <= 3.8e-282)
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * tan(t)) / Float64(-ew))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -8.5e-191], t$95$1, If[LessEqual[ew, 3.8e-282], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -8.49999999999999954e-191 or 3.79999999999999992e-282 < ew

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   7. Simplified 93.1%

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

   if -8.49999999999999954e-191 < ew < 3.79999999999999992e-282

   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 ew around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 80.4

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

   5. Simplified 80.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -8.5 \cdot 10^{-191}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{elif}\;ew \leq 3.8 \cdot 10^{-282}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}{ew}, -\cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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)))))))