Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.7% → 95.4%

Time: 26.1s

Alternatives: 7

Speedup: 40.5×

• 33.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)))))

Initial Program: 24.7% accurate, 1.0× speedup

Math

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

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)))))

Alternative 1: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale} \cdot a\\ t_1 := \mathsf{PI}\left(\right) \cdot \frac{angle}{180}\\ t_2 := \sin t\_1\\ t_3 := \cos t\_1\\ t_4 := \frac{\frac{t\_3 \cdot \left(t\_2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)\right)}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_4 \cdot t\_4 - \frac{\frac{{\left(t\_2 \cdot b\right)}^{2} + {\left(t\_3 \cdot a\right)}^{2}}{y-scale}}{y-scale} \cdot \left(\frac{\frac{{\left(t\_3 \cdot b\right)}^{2} + {\left(t\_2 \cdot a\right)}^{2}}{x-scale}}{x-scale} \cdot 4\right) \leq 0:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot -4\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ b (* y-scale x-scale)) a))
        (t_1 (* (PI) (/ angle 180.0)))
        (t_2 (sin t_1))
        (t_3 (cos t_1))
        (t_4
         (/
          (/ (* t_3 (* t_2 (* (- (pow b 2.0) (pow a 2.0)) 2.0))) x-scale)
          y-scale)))
   (if (<=
        (-
         (* t_4 t_4)
         (*
          (/ (/ (+ (pow (* t_2 b) 2.0) (pow (* t_3 a) 2.0)) y-scale) y-scale)
          (*
           (/ (/ (+ (pow (* t_3 b) 2.0) (pow (* t_2 a) 2.0)) x-scale) x-scale)
           4.0)))
        0.0)
     (* -4.0 (pow (* (/ b x-scale) (/ a y-scale)) 2.0))
     (* (* t_0 -4.0) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale))) < 0.0

   1. Initial program 68.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 92.7%

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

   11. Applied rewrites 99.6%

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

   if 0.0 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)))

   1. Initial program 0.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   7. Applied rewrites 67.3%

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

   9. Applied rewrites 95.1%

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

   11. Applied rewrites 95.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)\right)}{x-scale}}{y-scale} - \frac{\frac{{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right)}^{2} + {\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot a\right)}^{2}}{y-scale}}{y-scale} \cdot \left(\frac{\frac{{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right)}^{2} + {\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot a\right)}^{2}}{x-scale}}{x-scale} \cdot 4\right) \leq 0:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{b}{y-scale \cdot x-scale} \cdot a\right) \cdot -4\right) \cdot \left(\frac{b}{y-scale \cdot x-scale} \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.5% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale} \cdot a\\ \left(t\_0 \cdot -4\right) \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ b (* y-scale x-scale)) a))) (* (* t_0 -4.0) t_0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / (y_45_scale * x_45_scale)) * a;
	return (t_0 * -4.0) * t_0;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = (b / (y_45scale * x_45scale)) * a
    code = (t_0 * (-4.0d0)) * t_0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / (y_45_scale * x_45_scale)) * a;
	return (t_0 * -4.0) * t_0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b / (y_45_scale * x_45_scale)) * a
	return (t_0 * -4.0) * t_0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b / Float64(y_45_scale * x_45_scale)) * a)
	return Float64(Float64(t_0 * -4.0) * t_0)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b / (y_45_scale * x_45_scale)) * a;
	tmp = (t_0 * -4.0) * t_0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, N[(N[(t$95$0 * -4.0), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 22.4%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 94.3%

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

11. Applied rewrites 94.3%

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

12. Final simplification 94.3%

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

Alternative 3: 91.8% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ \left(\left(\left(t\_0 \cdot a\right) \cdot t\_0\right) \cdot a\right) \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale)))) (* (* (* (* t_0 a) t_0) a) -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	return (((t_0 * a) * t_0) * a) * -4.0;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = b / (y_45scale * x_45scale)
    code = (((t_0 * a) * t_0) * a) * (-4.0d0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	return (((t_0 * a) * t_0) * a) * -4.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	return (((t_0 * a) * t_0) * a) * -4.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale))
	return Float64(Float64(Float64(Float64(t_0 * a) * t_0) * a) * -4.0)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	tmp = (((t_0 * a) * t_0) * a) * -4.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(t$95$0 * a), $MachinePrecision] * t$95$0), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Initial program 22.4%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 94.3%

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

11. Applied rewrites 89.1%

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

12. Final simplification 89.1%

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

Alternative 4: 79.3% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ \left(\left(\left(a \cdot a\right) \cdot t\_0\right) \cdot t\_0\right) \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale)))) (* (* (* (* a a) t_0) t_0) -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	return (((a * a) * t_0) * t_0) * -4.0;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = b / (y_45scale * x_45scale)
    code = (((a * a) * t_0) * t_0) * (-4.0d0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	return (((a * a) * t_0) * t_0) * -4.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	return (((a * a) * t_0) * t_0) * -4.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale))
	return Float64(Float64(Float64(Float64(a * a) * t_0) * t_0) * -4.0)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	tmp = (((a * a) * t_0) * t_0) * -4.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(a * a), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Initial program 22.4%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 94.3%

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

11. Applied rewrites 79.1%

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

12. Final simplification 79.1%

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

Alternative 5: 75.9% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ \left(t\_0 \cdot t\_0\right) \cdot \left(\left(a \cdot a\right) \cdot -4\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale)))) (* (* t_0 t_0) (* (* a a) -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	return (t_0 * t_0) * ((a * a) * -4.0);
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = b / (y_45scale * x_45scale)
    code = (t_0 * t_0) * ((a * a) * (-4.0d0))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	return (t_0 * t_0) * ((a * a) * -4.0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	return (t_0 * t_0) * ((a * a) * -4.0)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale))
	return Float64(Float64(t_0 * t_0) * Float64(Float64(a * a) * -4.0))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	tmp = (t_0 * t_0) * ((a * a) * -4.0);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 22.4%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 72.0%

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

8. Final simplification 72.0%

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

Alternative 6: 61.7% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(\left(-4 \cdot a\right) \cdot a\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (/ (* b b) (* (* y-scale x-scale) (* y-scale x-scale))) (* (* -4.0 a) a)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * ((-4.0 * a) * a);
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((b * b) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (((-4.0d0) * a) * a)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * ((-4.0 * a) * a);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * ((-4.0 * a) * a)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(b * b) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * Float64(Float64(-4.0 * a) * a))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * ((-4.0 * a) * a);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(b * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.4%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 58.0%

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

10. Applied rewrites 58.0%

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

11. Final simplification 58.0%

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

Alternative 7: 61.7% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(\left(a \cdot a\right) \cdot -4\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (/ (* b b) (* (* y-scale x-scale) (* y-scale x-scale))) (* (* a a) -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * ((a * a) * -4.0);
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((b * b) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * ((a * a) * (-4.0d0))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * ((a * a) * -4.0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * ((a * a) * -4.0)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(b * b) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * Float64(Float64(a * a) * -4.0))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((b * b) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * ((a * a) * -4.0);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(b * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.4%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \]
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-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 58.0%

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

9. Final simplification 58.0%

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

Reproduce

herbie shell --seed 2024278 
(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))))