Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 26.2% → 92.7%

Time: 25.8s

Alternatives: 9

Speedup: 40.5×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 3.284101232254139e+219
  2. b = -2.6284331920889544e+284
  3. angle = 1.897727289283392e+101
  4. x-scale = 1.6399242950528922e-277
  5. y-scale = -2.1421258921529646e-198

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 26.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 92.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ -4 \cdot e^{\mathsf{fma}\left(2, \log b\_m, 2 \cdot \log a\_m\right) - 2 \cdot \log \left(x-scale\_m \cdot y-scale\_m\right)} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  -4.0
  (exp
   (-
    (fma 2.0 (log b_m) (* 2.0 (log a_m)))
    (* 2.0 (log (* x-scale_m y-scale_m)))))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * exp((fma(2.0, log(b_m), (2.0 * log(a_m))) - (2.0 * log((x_45_scale_m * y_45_scale_m)))));
}

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(-4.0 * exp(Float64(fma(2.0, log(b_m), Float64(2.0 * log(a_m))) - Float64(2.0 * log(Float64(x_45_scale_m * y_45_scale_m))))))
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(-4.0 * N[Exp[N[(N[(2.0 * N[Log[b$95$m], $MachinePrecision] + N[(2.0 * N[Log[a$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Log[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   16. lower-*.f64 51.7

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 51.1%

   \[\leadsto -4 \cdot \color{blue}{\frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 63.2%

   \[\leadsto -4 \cdot \frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 13.2%

    \[\leadsto -4 \cdot e^{\mathsf{fma}\left(2, \log b, 2 \cdot \log a\right) - \log \left(x-scale \cdot y-scale\right) \cdot 2} \]

12. Final simplification 13.2%

    \[\leadsto -4 \cdot e^{\mathsf{fma}\left(2, \log b, 2 \cdot \log a\right) - 2 \cdot \log \left(x-scale \cdot y-scale\right)} \]
13. Add Preprocessing

Alternative 2: 76.7% accurate, 32.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := x-scale\_m \cdot \left(y-scale\_m \cdot \left(x-scale\_m \cdot y-scale\_m\right)\right)\\ t_1 := b\_m \cdot \frac{a\_m \cdot \left(a\_m \cdot \left(b\_m \cdot -4\right)\right)}{t\_0}\\ \mathbf{if}\;b\_m \leq 3 \cdot 10^{-150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b\_m \leq 7.2 \cdot 10^{+169}:\\ \;\;\;\;\left(a\_m \cdot -4\right) \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \frac{a\_m}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* x-scale_m (* y-scale_m (* x-scale_m y-scale_m))))
        (t_1 (* b_m (/ (* a_m (* a_m (* b_m -4.0))) t_0))))
   (if (<= b_m 3e-150)
     t_1
     (if (<= b_m 7.2e+169) (* (* a_m -4.0) (* (* b_m b_m) (/ a_m t_0))) t_1))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = x_45_scale_m * (y_45_scale_m * (x_45_scale_m * y_45_scale_m));
	double t_1 = b_m * ((a_m * (a_m * (b_m * -4.0))) / t_0);
	double tmp;
	if (b_m <= 3e-150) {
		tmp = t_1;
	} else if (b_m <= 7.2e+169) {
		tmp = (a_m * -4.0) * ((b_m * b_m) * (a_m / t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_45scale_m * (y_45scale_m * (x_45scale_m * y_45scale_m))
    t_1 = b_m * ((a_m * (a_m * (b_m * (-4.0d0)))) / t_0)
    if (b_m <= 3d-150) then
        tmp = t_1
    else if (b_m <= 7.2d+169) then
        tmp = (a_m * (-4.0d0)) * ((b_m * b_m) * (a_m / t_0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = x_45_scale_m * (y_45_scale_m * (x_45_scale_m * y_45_scale_m));
	double t_1 = b_m * ((a_m * (a_m * (b_m * -4.0))) / t_0);
	double tmp;
	if (b_m <= 3e-150) {
		tmp = t_1;
	} else if (b_m <= 7.2e+169) {
		tmp = (a_m * -4.0) * ((b_m * b_m) * (a_m / t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = x_45_scale_m * (y_45_scale_m * (x_45_scale_m * y_45_scale_m))
	t_1 = b_m * ((a_m * (a_m * (b_m * -4.0))) / t_0)
	tmp = 0
	if b_m <= 3e-150:
		tmp = t_1
	elif b_m <= 7.2e+169:
		tmp = (a_m * -4.0) * ((b_m * b_m) * (a_m / t_0))
	else:
		tmp = t_1
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(x_45_scale_m * y_45_scale_m)))
	t_1 = Float64(b_m * Float64(Float64(a_m * Float64(a_m * Float64(b_m * -4.0))) / t_0))
	tmp = 0.0
	if (b_m <= 3e-150)
		tmp = t_1;
	elseif (b_m <= 7.2e+169)
		tmp = Float64(Float64(a_m * -4.0) * Float64(Float64(b_m * b_m) * Float64(a_m / t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = x_45_scale_m * (y_45_scale_m * (x_45_scale_m * y_45_scale_m));
	t_1 = b_m * ((a_m * (a_m * (b_m * -4.0))) / t_0);
	tmp = 0.0;
	if (b_m <= 3e-150)
		tmp = t_1;
	elseif (b_m <= 7.2e+169)
		tmp = (a_m * -4.0) * ((b_m * b_m) * (a_m / t_0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b$95$m * N[(N[(a$95$m * N[(a$95$m * N[(b$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 3e-150], t$95$1, If[LessEqual[b$95$m, 7.2e+169], N[(N[(a$95$m * -4.0), $MachinePrecision] * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(a$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.0000000000000002e-150 or 7.20000000000000019e169 < b

   1. Initial program 24.5%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 49.5

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.3%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.1%

      \[\leadsto \frac{\left(\left(a \cdot b\right) \cdot -4\right) \cdot b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.9%

       \[\leadsto b \cdot \color{blue}{\frac{a \cdot \left(a \cdot \left(b \cdot -4\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}} \]

   if 3.0000000000000002e-150 < b < 7.20000000000000019e169

   1. Initial program 21.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 57.2

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.6%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.8%

      \[\leadsto \frac{\left(\left(a \cdot b\right) \cdot -4\right) \cdot b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.7%

       \[\leadsto \left(a \cdot -4\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \frac{a}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 78.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := b\_m \cdot \frac{a\_m \cdot \left(a\_m \cdot \left(b\_m \cdot -4\right)\right)}{x-scale\_m \cdot \left(y-scale\_m \cdot \left(x-scale\_m \cdot y-scale\_m\right)\right)}\\ \mathbf{if}\;b\_m \leq 2.7 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_m \leq 2 \cdot 10^{+239}:\\ \;\;\;\;a\_m \cdot \left(\left(-4 \cdot \left(b\_m \cdot a\_m\right)\right) \cdot \frac{b\_m}{y-scale\_m \cdot \left(x-scale\_m \cdot \left(x-scale\_m \cdot y-scale\_m\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0
         (*
          b_m
          (/
           (* a_m (* a_m (* b_m -4.0)))
           (* x-scale_m (* y-scale_m (* x-scale_m y-scale_m)))))))
   (if (<= b_m 2.7e-150)
     t_0
     (if (<= b_m 2e+239)
       (*
        a_m
        (*
         (* -4.0 (* b_m a_m))
         (/ b_m (* y-scale_m (* x-scale_m (* x-scale_m y-scale_m))))))
       t_0))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * ((a_m * (a_m * (b_m * -4.0))) / (x_45_scale_m * (y_45_scale_m * (x_45_scale_m * y_45_scale_m))));
	double tmp;
	if (b_m <= 2.7e-150) {
		tmp = t_0;
	} else if (b_m <= 2e+239) {
		tmp = a_m * ((-4.0 * (b_m * a_m)) * (b_m / (y_45_scale_m * (x_45_scale_m * (x_45_scale_m * y_45_scale_m)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b_m * ((a_m * (a_m * (b_m * (-4.0d0)))) / (x_45scale_m * (y_45scale_m * (x_45scale_m * y_45scale_m))))
    if (b_m <= 2.7d-150) then
        tmp = t_0
    else if (b_m <= 2d+239) then
        tmp = a_m * (((-4.0d0) * (b_m * a_m)) * (b_m / (y_45scale_m * (x_45scale_m * (x_45scale_m * y_45scale_m)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * ((a_m * (a_m * (b_m * -4.0))) / (x_45_scale_m * (y_45_scale_m * (x_45_scale_m * y_45_scale_m))));
	double tmp;
	if (b_m <= 2.7e-150) {
		tmp = t_0;
	} else if (b_m <= 2e+239) {
		tmp = a_m * ((-4.0 * (b_m * a_m)) * (b_m / (y_45_scale_m * (x_45_scale_m * (x_45_scale_m * y_45_scale_m)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = b_m * ((a_m * (a_m * (b_m * -4.0))) / (x_45_scale_m * (y_45_scale_m * (x_45_scale_m * y_45_scale_m))))
	tmp = 0
	if b_m <= 2.7e-150:
		tmp = t_0
	elif b_m <= 2e+239:
		tmp = a_m * ((-4.0 * (b_m * a_m)) * (b_m / (y_45_scale_m * (x_45_scale_m * (x_45_scale_m * y_45_scale_m)))))
	else:
		tmp = t_0
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(Float64(a_m * Float64(a_m * Float64(b_m * -4.0))) / Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(x_45_scale_m * y_45_scale_m)))))
	tmp = 0.0
	if (b_m <= 2.7e-150)
		tmp = t_0;
	elseif (b_m <= 2e+239)
		tmp = Float64(a_m * Float64(Float64(-4.0 * Float64(b_m * a_m)) * Float64(b_m / Float64(y_45_scale_m * Float64(x_45_scale_m * Float64(x_45_scale_m * y_45_scale_m))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = b_m * ((a_m * (a_m * (b_m * -4.0))) / (x_45_scale_m * (y_45_scale_m * (x_45_scale_m * y_45_scale_m))));
	tmp = 0.0;
	if (b_m <= 2.7e-150)
		tmp = t_0;
	elseif (b_m <= 2e+239)
		tmp = a_m * ((-4.0 * (b_m * a_m)) * (b_m / (y_45_scale_m * (x_45_scale_m * (x_45_scale_m * y_45_scale_m)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(N[(a$95$m * N[(a$95$m * N[(b$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 2.7e-150], t$95$0, If[LessEqual[b$95$m, 2e+239], N[(a$95$m * N[(N[(-4.0 * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / N[(y$45$scale$95$m * N[(x$45$scale$95$m * N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000001e-150 or 1.99999999999999998e239 < b

   1. Initial program 26.7%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 51.5

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.6%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 81.2%

      \[\leadsto \frac{\left(\left(a \cdot b\right) \cdot -4\right) \cdot b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.6%

       \[\leadsto b \cdot \color{blue}{\frac{a \cdot \left(a \cdot \left(b \cdot -4\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}} \]

   if 2.7000000000000001e-150 < b < 1.99999999999999998e239

   1. Initial program 17.7%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 52.1

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.8%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]

   9. Applied rewrites 80.1%

      \[\leadsto a \cdot \color{blue}{\left(\left(\left(a \cdot b\right) \cdot -4\right) \cdot \frac{b}{y-scale \cdot \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-150}:\\ \;\;\;\;b \cdot \frac{a \cdot \left(a \cdot \left(b \cdot -4\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+239}:\\ \;\;\;\;a \cdot \left(\left(-4 \cdot \left(b \cdot a\right)\right) \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{a \cdot \left(a \cdot \left(b \cdot -4\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := a\_m \cdot \left(b\_m \cdot -4\right)\\ \mathbf{if}\;b\_m \leq 1.3 \cdot 10^{+239}:\\ \;\;\;\;\left(b\_m \cdot \frac{t\_0}{x-scale\_m \cdot y-scale\_m}\right) \cdot \frac{a\_m}{x-scale\_m \cdot y-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b\_m \cdot \left(a\_m \cdot t\_0\right)}{x-scale\_m \cdot y-scale\_m}}{x-scale\_m \cdot y-scale\_m}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* a_m (* b_m -4.0))))
   (if (<= b_m 1.3e+239)
     (*
      (* b_m (/ t_0 (* x-scale_m y-scale_m)))
      (/ a_m (* x-scale_m y-scale_m)))
     (/
      (/ (* b_m (* a_m t_0)) (* x-scale_m y-scale_m))
      (* x-scale_m y-scale_m)))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = a_m * (b_m * -4.0);
	double tmp;
	if (b_m <= 1.3e+239) {
		tmp = (b_m * (t_0 / (x_45_scale_m * y_45_scale_m))) * (a_m / (x_45_scale_m * y_45_scale_m));
	} else {
		tmp = ((b_m * (a_m * t_0)) / (x_45_scale_m * y_45_scale_m)) / (x_45_scale_m * y_45_scale_m);
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (b_m * (-4.0d0))
    if (b_m <= 1.3d+239) then
        tmp = (b_m * (t_0 / (x_45scale_m * y_45scale_m))) * (a_m / (x_45scale_m * y_45scale_m))
    else
        tmp = ((b_m * (a_m * t_0)) / (x_45scale_m * y_45scale_m)) / (x_45scale_m * y_45scale_m)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = a_m * (b_m * -4.0);
	double tmp;
	if (b_m <= 1.3e+239) {
		tmp = (b_m * (t_0 / (x_45_scale_m * y_45_scale_m))) * (a_m / (x_45_scale_m * y_45_scale_m));
	} else {
		tmp = ((b_m * (a_m * t_0)) / (x_45_scale_m * y_45_scale_m)) / (x_45_scale_m * y_45_scale_m);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = a_m * (b_m * -4.0)
	tmp = 0
	if b_m <= 1.3e+239:
		tmp = (b_m * (t_0 / (x_45_scale_m * y_45_scale_m))) * (a_m / (x_45_scale_m * y_45_scale_m))
	else:
		tmp = ((b_m * (a_m * t_0)) / (x_45_scale_m * y_45_scale_m)) / (x_45_scale_m * y_45_scale_m)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(a_m * Float64(b_m * -4.0))
	tmp = 0.0
	if (b_m <= 1.3e+239)
		tmp = Float64(Float64(b_m * Float64(t_0 / Float64(x_45_scale_m * y_45_scale_m))) * Float64(a_m / Float64(x_45_scale_m * y_45_scale_m)));
	else
		tmp = Float64(Float64(Float64(b_m * Float64(a_m * t_0)) / Float64(x_45_scale_m * y_45_scale_m)) / Float64(x_45_scale_m * y_45_scale_m));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = a_m * (b_m * -4.0);
	tmp = 0.0;
	if (b_m <= 1.3e+239)
		tmp = (b_m * (t_0 / (x_45_scale_m * y_45_scale_m))) * (a_m / (x_45_scale_m * y_45_scale_m));
	else
		tmp = ((b_m * (a_m * t_0)) / (x_45_scale_m * y_45_scale_m)) / (x_45_scale_m * y_45_scale_m);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(a$95$m * N[(b$95$m * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.3e+239], N[(N[(b$95$m * N[(t$95$0 / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a$95$m / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m * N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3000000000000001e239

   1. Initial program 25.9%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 52.5

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.8%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 84.7%

      \[\leadsto \frac{\left(\left(a \cdot b\right) \cdot -4\right) \cdot b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.5%

       \[\leadsto \left(b \cdot \frac{a \cdot \left(b \cdot -4\right)}{x-scale \cdot y-scale}\right) \cdot \frac{\color{blue}{a}}{x-scale \cdot y-scale} \]

   if 1.3000000000000001e239 < b

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 43.7

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(a \cdot b\right) \cdot -4\right) \cdot b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale} \]

   11. Applied rewrites 78.5%

       \[\leadsto \frac{\frac{b \cdot \left(a \cdot \left(a \cdot \left(b \cdot -4\right)\right)\right)}{x-scale \cdot y-scale}}{\color{blue}{x-scale \cdot y-scale}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.8% accurate, 35.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \left(b\_m \cdot \frac{a\_m \cdot \left(b\_m \cdot -4\right)}{x-scale\_m \cdot y-scale\_m}\right) \cdot \frac{a\_m}{x-scale\_m \cdot y-scale\_m} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  (* b_m (/ (* a_m (* b_m -4.0)) (* x-scale_m y-scale_m)))
  (/ a_m (* x-scale_m y-scale_m))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (b_m * ((a_m * (b_m * -4.0)) / (x_45_scale_m * y_45_scale_m))) * (a_m / (x_45_scale_m * y_45_scale_m));
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (b_m * ((a_m * (b_m * (-4.0d0))) / (x_45scale_m * y_45scale_m))) * (a_m / (x_45scale_m * y_45scale_m))
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (b_m * ((a_m * (b_m * -4.0)) / (x_45_scale_m * y_45_scale_m))) * (a_m / (x_45_scale_m * y_45_scale_m));
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return (b_m * ((a_m * (b_m * -4.0)) / (x_45_scale_m * y_45_scale_m))) * (a_m / (x_45_scale_m * y_45_scale_m))

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(Float64(b_m * Float64(Float64(a_m * Float64(b_m * -4.0)) / Float64(x_45_scale_m * y_45_scale_m))) * Float64(a_m / Float64(x_45_scale_m * y_45_scale_m)))
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = (b_m * ((a_m * (b_m * -4.0)) / (x_45_scale_m * y_45_scale_m))) * (a_m / (x_45_scale_m * y_45_scale_m));
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(N[(b$95$m * N[(N[(a$95$m * N[(b$95$m * -4.0), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a$95$m / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   16. lower-*.f64 51.7

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 77.4%

   \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]
8. Step-by-step derivation

9. Applied rewrites 81.9%

   \[\leadsto \frac{\left(\left(a \cdot b\right) \cdot -4\right) \cdot b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale} \]
10. Step-by-step derivation

11. Applied rewrites 89.5%

    \[\leadsto \left(b \cdot \frac{a \cdot \left(b \cdot -4\right)}{x-scale \cdot y-scale}\right) \cdot \frac{\color{blue}{a}}{x-scale \cdot y-scale} \]
12. Add Preprocessing

Alternative 6: 89.5% accurate, 35.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \left(a\_m \cdot \left(b\_m \cdot -4\right)\right) \cdot \left(\frac{a\_m}{x-scale\_m \cdot y-scale\_m} \cdot \frac{b\_m}{x-scale\_m \cdot y-scale\_m}\right) \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  (* a_m (* b_m -4.0))
  (* (/ a_m (* x-scale_m y-scale_m)) (/ b_m (* x-scale_m y-scale_m)))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (a_m * (b_m * -4.0)) * ((a_m / (x_45_scale_m * y_45_scale_m)) * (b_m / (x_45_scale_m * y_45_scale_m)));
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (a_m * (b_m * (-4.0d0))) * ((a_m / (x_45scale_m * y_45scale_m)) * (b_m / (x_45scale_m * y_45scale_m)))
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (a_m * (b_m * -4.0)) * ((a_m / (x_45_scale_m * y_45_scale_m)) * (b_m / (x_45_scale_m * y_45_scale_m)));
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return (a_m * (b_m * -4.0)) * ((a_m / (x_45_scale_m * y_45_scale_m)) * (b_m / (x_45_scale_m * y_45_scale_m)))

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(Float64(a_m * Float64(b_m * -4.0)) * Float64(Float64(a_m / Float64(x_45_scale_m * y_45_scale_m)) * Float64(b_m / Float64(x_45_scale_m * y_45_scale_m))))
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = (a_m * (b_m * -4.0)) * ((a_m / (x_45_scale_m * y_45_scale_m)) * (b_m / (x_45_scale_m * y_45_scale_m)));
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(N[(a$95$m * N[(b$95$m * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a$95$m / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   16. lower-*.f64 51.7

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 77.4%

   \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]
8. Step-by-step derivation

9. Applied rewrites 81.9%

   \[\leadsto \frac{\left(\left(a \cdot b\right) \cdot -4\right) \cdot b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale} \]
10. Step-by-step derivation

11. Applied rewrites 88.1%

    \[\leadsto \left(a \cdot \left(b \cdot -4\right)\right) \cdot \color{blue}{\left(\frac{b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale}\right)} \]

12. Final simplification 88.1%

    \[\leadsto \left(a \cdot \left(b \cdot -4\right)\right) \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \frac{b}{x-scale \cdot y-scale}\right) \]
13. Add Preprocessing

Alternative 7: 78.1% accurate, 40.5× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ a\_m \cdot \left(\left(-4 \cdot \left(b\_m \cdot a\_m\right)\right) \cdot \frac{b\_m}{y-scale\_m \cdot \left(x-scale\_m \cdot \left(x-scale\_m \cdot y-scale\_m\right)\right)}\right) \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  a_m
  (*
   (* -4.0 (* b_m a_m))
   (/ b_m (* y-scale_m (* x-scale_m (* x-scale_m y-scale_m)))))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return a_m * ((-4.0 * (b_m * a_m)) * (b_m / (y_45_scale_m * (x_45_scale_m * (x_45_scale_m * y_45_scale_m)))));
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = a_m * (((-4.0d0) * (b_m * a_m)) * (b_m / (y_45scale_m * (x_45scale_m * (x_45scale_m * y_45scale_m)))))
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return a_m * ((-4.0 * (b_m * a_m)) * (b_m / (y_45_scale_m * (x_45_scale_m * (x_45_scale_m * y_45_scale_m)))));
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return a_m * ((-4.0 * (b_m * a_m)) * (b_m / (y_45_scale_m * (x_45_scale_m * (x_45_scale_m * y_45_scale_m)))))

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(a_m * Float64(Float64(-4.0 * Float64(b_m * a_m)) * Float64(b_m / Float64(y_45_scale_m * Float64(x_45_scale_m * Float64(x_45_scale_m * y_45_scale_m))))))
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = a_m * ((-4.0 * (b_m * a_m)) * (b_m / (y_45_scale_m * (x_45_scale_m * (x_45_scale_m * y_45_scale_m)))));
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(a$95$m * N[(N[(-4.0 * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / N[(y$45$scale$95$m * N[(x$45$scale$95$m * N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   16. lower-*.f64 51.7

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 77.4%

   \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{a}{x-scale \cdot y-scale}} \]

9. Applied rewrites 74.3%

   \[\leadsto a \cdot \color{blue}{\left(\left(\left(a \cdot b\right) \cdot -4\right) \cdot \frac{b}{y-scale \cdot \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right)}\right)} \]

10. Final simplification 74.3%

    \[\leadsto a \cdot \left(\left(-4 \cdot \left(b \cdot a\right)\right) \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right) \]
11. Add Preprocessing

Alternative 8: 75.2% accurate, 40.5× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ -4 \cdot \frac{a\_m \cdot \left(b\_m \cdot \left(b\_m \cdot a\_m\right)\right)}{\left(x-scale\_m \cdot y-scale\_m\right) \cdot \left(x-scale\_m \cdot y-scale\_m\right)} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  -4.0
  (/
   (* a_m (* b_m (* b_m a_m)))
   (* (* x-scale_m y-scale_m) (* x-scale_m y-scale_m)))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * ((a_m * (b_m * (b_m * a_m))) / ((x_45_scale_m * y_45_scale_m) * (x_45_scale_m * y_45_scale_m)));
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (-4.0d0) * ((a_m * (b_m * (b_m * a_m))) / ((x_45scale_m * y_45scale_m) * (x_45scale_m * y_45scale_m)))
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * ((a_m * (b_m * (b_m * a_m))) / ((x_45_scale_m * y_45_scale_m) * (x_45_scale_m * y_45_scale_m)));
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return -4.0 * ((a_m * (b_m * (b_m * a_m))) / ((x_45_scale_m * y_45_scale_m) * (x_45_scale_m * y_45_scale_m)))

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(-4.0 * Float64(Float64(a_m * Float64(b_m * Float64(b_m * a_m))) / Float64(Float64(x_45_scale_m * y_45_scale_m) * Float64(x_45_scale_m * y_45_scale_m))))
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = -4.0 * ((a_m * (b_m * (b_m * a_m))) / ((x_45_scale_m * y_45_scale_m) * (x_45_scale_m * y_45_scale_m)));
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(-4.0 * N[(N[(a$95$m * N[(b$95$m * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   16. lower-*.f64 51.7

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 51.1%

   \[\leadsto -4 \cdot \color{blue}{\frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 63.2%

   \[\leadsto -4 \cdot \frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 70.0%

    \[\leadsto -4 \cdot \frac{\left(b \cdot \left(a \cdot b\right)\right) \cdot a}{\color{blue}{\left(x-scale \cdot y-scale\right)} \cdot \left(x-scale \cdot y-scale\right)} \]

12. Final simplification 70.0%

    \[\leadsto -4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot a\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \]
13. Add Preprocessing

Alternative 9: 67.8% accurate, 40.5× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ -4 \cdot \frac{b\_m \cdot \left(b\_m \cdot \left(a\_m \cdot a\_m\right)\right)}{\left(x-scale\_m \cdot y-scale\_m\right) \cdot \left(x-scale\_m \cdot y-scale\_m\right)} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  -4.0
  (/
   (* b_m (* b_m (* a_m a_m)))
   (* (* x-scale_m y-scale_m) (* x-scale_m y-scale_m)))))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * ((b_m * (b_m * (a_m * a_m))) / ((x_45_scale_m * y_45_scale_m) * (x_45_scale_m * y_45_scale_m)));
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (-4.0d0) * ((b_m * (b_m * (a_m * a_m))) / ((x_45scale_m * y_45scale_m) * (x_45scale_m * y_45scale_m)))
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return -4.0 * ((b_m * (b_m * (a_m * a_m))) / ((x_45_scale_m * y_45_scale_m) * (x_45_scale_m * y_45_scale_m)));
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return -4.0 * ((b_m * (b_m * (a_m * a_m))) / ((x_45_scale_m * y_45_scale_m) * (x_45_scale_m * y_45_scale_m)))

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(-4.0 * Float64(Float64(b_m * Float64(b_m * Float64(a_m * a_m))) / Float64(Float64(x_45_scale_m * y_45_scale_m) * Float64(x_45_scale_m * y_45_scale_m))))
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = -4.0 * ((b_m * (b_m * (a_m * a_m))) / ((x_45_scale_m * y_45_scale_m) * (x_45_scale_m * y_45_scale_m)));
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(-4.0 * N[(N[(b$95$m * N[(b$95$m * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   16. lower-*.f64 51.7

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 51.1%

   \[\leadsto -4 \cdot \color{blue}{\frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 63.2%

   \[\leadsto -4 \cdot \frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale))))