b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 35.3%

Time: 44.2s

Alternatives: 4

Speedup: 484.7×

• could not determine a ground truth

  1. a = 1.2414043078309434e+177
  2. b = -4.341614761951697e-296
  3. angle = -8.265104169209068e-119
  4. x-scale = -3.597992818941878e+302
  5. y-scale = -1.0465563771021355e-146

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) - \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

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
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (-
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

Initial Program: 0.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) - \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

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
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (-
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

Alternative 1: 35.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{b\_m \cdot b\_m}{x-scale\_m}\\ t_1 := \left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := {\cos t\_1}^{2}\\ t_3 := \frac{t\_2}{x-scale\_m}\\ t_4 := {\sin t\_1}^{2}\\ \mathbf{if}\;b\_m \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(\sqrt{8} \cdot y-scale\right) \cdot x-scale\_m\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{b\_m \cdot b\_m}{y-scale}, \frac{t\_4}{y-scale}, t\_3 \cdot t\_0\right) - \sqrt{\mathsf{fma}\left(\frac{4}{y-scale \cdot y-scale}, \frac{{b\_m}^{4} \cdot t\_2}{x-scale\_m} \cdot \frac{t\_4}{x-scale\_m}, {\left(\mathsf{fma}\left(t\_3, t\_0, \left(b\_m \cdot b\_m\right) \cdot \frac{t\_4}{\left(-y-scale\right) \cdot y-scale}\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (let* ((t_0 (/ (* b_m b_m) x-scale_m))
        (t_1 (* (* (PI) angle) 0.005555555555555556))
        (t_2 (pow (cos t_1) 2.0))
        (t_3 (/ t_2 x-scale_m))
        (t_4 (pow (sin t_1) 2.0)))
   (if (<= b_m 4.2e-204)
     (*
      (* 0.25 (* (* (sqrt 8.0) y-scale) x-scale_m))
      (sqrt
       (-
        (fma (/ (* b_m b_m) y-scale) (/ t_4 y-scale) (* t_3 t_0))
        (sqrt
         (fma
          (/ 4.0 (* y-scale y-scale))
          (* (/ (* (pow b_m 4.0) t_2) x-scale_m) (/ t_4 x-scale_m))
          (pow
           (fma t_3 t_0 (* (* b_m b_m) (/ t_4 (* (- y-scale) y-scale))))
           2.0))))))
     (* a_m x-scale_m))))

Derivation

1. Split input into 2 regimes

2. if b < 4.20000000000000018e-204

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}\right)} \]

   5. Applied rewrites 13.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(\sqrt{8} \cdot y-scale\right) \cdot x-scale\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{b \cdot b}{y-scale}, \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale}, \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \frac{b \cdot b}{x-scale}\right) - \sqrt{\mathsf{fma}\left(\frac{4}{y-scale \cdot y-scale}, \frac{{b}^{4} \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}, {\left(\mathsf{fma}\left(\frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}, \frac{b \cdot b}{x-scale}, \left(-b \cdot b\right) \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}\right)\right)}^{2}\right)}}} \]

   if 4.20000000000000018e-204 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right)} \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\color{blue}{\left(x-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\left(x-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 26.7

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 26.9%

      \[\leadsto \left(\left(0.25 \cdot a\right) \cdot x-scale\right) \cdot \color{blue}{4} \]

   8. Taylor expanded in a around 0

      \[\leadsto a \cdot \color{blue}{x-scale} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.9%

       \[\leadsto a \cdot \color{blue}{x-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(\sqrt{8} \cdot y-scale\right) \cdot x-scale\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{b \cdot b}{y-scale}, \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale}, \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \frac{b \cdot b}{x-scale}\right) - \sqrt{\mathsf{fma}\left(\frac{4}{y-scale \cdot y-scale}, \frac{{b}^{4} \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}, {\left(\mathsf{fma}\left(\frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}, \frac{b \cdot b}{x-scale}, \left(b \cdot b\right) \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{\left(-y-scale\right) \cdot y-scale}\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 35.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\\ t_1 := {\cos t\_0}^{2}\\ t_2 := {\sin t\_0}^{2}\\ \mathbf{if}\;b\_m \leq 3.85 \cdot 10^{-205}:\\ \;\;\;\;\left(0.25 \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right) \cdot \sqrt{\frac{t\_2}{y-scale \cdot y-scale} - 0.5 \cdot \frac{\frac{t\_1 \cdot t\_2}{y-scale \cdot y-scale} \cdot 2}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle (PI))))
        (t_1 (pow (cos t_0) 2.0))
        (t_2 (pow (sin t_0) 2.0)))
   (if (<= b_m 3.85e-205)
     (*
      (* 0.25 (* b_m (* x-scale_m (* y-scale (sqrt 8.0)))))
      (sqrt
       (-
        (/ t_2 (* y-scale y-scale))
        (* 0.5 (/ (* (/ (* t_1 t_2) (* y-scale y-scale)) 2.0) t_1)))))
     (* a_m x-scale_m))))

Derivation

1. Split input into 2 regimes

2. if b < 3.85e-205

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}\right)} \]

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(\left(\sqrt{8} \cdot y-scale\right) \cdot x-scale\right) \cdot b\right)\right) \cdot \sqrt{\left(\frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale} + \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale \cdot x-scale}\right) - \sqrt{\mathsf{fma}\left(\frac{4}{y-scale \cdot y-scale}, \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}, {\left(\frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)}}} \]

   6. Taylor expanded in x-scale around 0

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \frac{1}{2} \cdot \frac{-2 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}}{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.2%

      \[\leadsto \left(0.25 \cdot \left(b \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale \cdot y-scale} - 0.5 \cdot \frac{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale \cdot y-scale} \cdot 2}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}}} \]

   if 3.85e-205 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right)} \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\color{blue}{\left(x-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\left(x-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 26.7

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 26.9%

      \[\leadsto \left(\left(0.25 \cdot a\right) \cdot x-scale\right) \cdot \color{blue}{4} \]

   8. Taylor expanded in a around 0

      \[\leadsto a \cdot \color{blue}{x-scale} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.9%

       \[\leadsto a \cdot \color{blue}{x-scale} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 33.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\\ t_1 := {\sin t\_0}^{2}\\ t_2 := {\cos t\_0}^{2}\\ \mathbf{if}\;b\_m \leq 1.8 \cdot 10^{-241}:\\ \;\;\;\;\left(0.25 \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(\frac{t\_2 \cdot t\_1}{x-scale\_m \cdot x-scale\_m} \cdot 2\right)}{t\_2}, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

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)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle (PI))))
        (t_1 (pow (sin t_0) 2.0))
        (t_2 (pow (cos t_0) 2.0)))
   (if (<= b_m 1.8e-241)
     (*
      (* 0.25 (* b_m (* x-scale_m (sqrt 8.0))))
      (sqrt
       (fma
        -0.5
        (/
         (*
          (* x-scale_m x-scale_m)
          (* (/ (* t_2 t_1) (* x-scale_m x-scale_m)) 2.0))
         t_2)
        t_1)))
     (* a_m x-scale_m))))

Derivation

1. Split input into 2 regimes

2. if b < 1.7999999999999999e-241

   1. Initial program 0.0%

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

   3. Taylor expanded in y-scale around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) - \frac{1}{2} \cdot \frac{-2 \cdot \left(\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) \cdot \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\left({b}^{2} - {a}^{2}\right)}^{2}\right)}{{x-scale}^{2}}}{\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}}}\right)} \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\sqrt{8} \cdot x-scale\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}, \mathsf{fma}\left(b \cdot b, {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}, -0.5 \cdot \frac{\mathsf{fma}\left(\frac{4}{x-scale}, \frac{\left({\left(\left(b + a\right) \cdot \left(b - a\right)\right)}^{2} \cdot {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}, \left(-2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}, \left(b \cdot b\right) \cdot {\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right)\right) \cdot \mathsf{fma}\left(\frac{a \cdot a}{x-scale}, \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}, \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \frac{b \cdot b}{x-scale}\right)\right)}{\mathsf{fma}\left(\frac{a \cdot a}{x-scale}, \frac{{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale}, \frac{{\cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \frac{b \cdot b}{x-scale}\right)}\right)\right)}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(\left(b \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\frac{-1}{2} \cdot \frac{{x-scale}^{2} \cdot \left(-2 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + 4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.4%

      \[\leadsto \left(0.25 \cdot \left(b \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(-0.5, \frac{\left(x-scale \cdot x-scale\right) \cdot \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale \cdot x-scale} \cdot 2\right)}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

   if 1.7999999999999999e-241 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right)} \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\color{blue}{\left(x-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\left(x-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

      8. lower-sqrt.f64 25.7

         \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

   5. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 25.8%

      \[\leadsto \left(\left(0.25 \cdot a\right) \cdot x-scale\right) \cdot \color{blue}{4} \]

   8. Taylor expanded in a around 0

      \[\leadsto a \cdot \color{blue}{x-scale} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.8%

       \[\leadsto a \cdot \color{blue}{x-scale} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 32.5% accurate, 484.7× speedup

Math

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

FPCore

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)
 :precision binary64
 (* a_m x-scale_m))

C

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) {
	return a_m * x_45_scale_m;
}

Fortran

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)
    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
    code = a_m * x_45scale_m
end function

Java

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) {
	return a_m * x_45_scale_m;
}

Python

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):
	return a_m * x_45_scale_m

Julia

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)
	return Float64(a_m * x_45_scale_m)
end

MATLAB

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)
	tmp = a_m * x_45_scale_m;
end

Wolfram

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_] := N[(a$95$m * x$45$scale$95$m), $MachinePrecision]

Derivation

1. Initial program 0.0%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right) \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot a\right)} \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\color{blue}{\left(x-scale \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \left(\left(x-scale \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{8}\right) \]

   8. lower-sqrt.f64 23.7

      \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right) \]

5. Applied rewrites 23.7%

   \[\leadsto \color{blue}{\left(0.25 \cdot a\right) \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 23.8%

   \[\leadsto \left(\left(0.25 \cdot a\right) \cdot x-scale\right) \cdot \color{blue}{4} \]

8. Taylor expanded in a around 0

   \[\leadsto a \cdot \color{blue}{x-scale} \]
9. Step-by-step derivation

10. Applied rewrites 23.8%

    \[\leadsto a \cdot \color{blue}{x-scale} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (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)) (sqrt (+ (pow (- (/ (/ (+ (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)) 2.0) (pow (/ (/ (* (* (* 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))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))