2-ancestry mixing, zero discriminant

Percentage Accurate: 78.8% → 98.8%

Time: 1.6s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary32 (cbrt (/ g (* 2.0 a))))

C

float code(float g, float a) {
	return cbrtf((g / (2.0f * a)));
}

Julia

function code(g, a)
	return cbrt(Float32(g / Float32(Float32(2.0) * a)))
end

Initial Program: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{2 \cdot a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary32 (cbrt (/ g (* 2.0 a))))

C

float code(float g, float a) {
	return cbrtf((g / (2.0f * a)));
}

Julia

function code(g, a)
	return cbrt(Float32(g / Float32(Float32(2.0) * a)))
end

Alternative 1: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{0.5 \cdot g}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary32 (/ (cbrt (* 0.5 g)) (cbrt a)))

C

float code(float g, float a) {
	return cbrtf((0.5f * g)) / cbrtf(a);
}

Julia

function code(g, a)
	return Float32(cbrt(Float32(Float32(0.5) * g)) / cbrt(a))
end

Derivation

1. Initial program 82.8%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cbrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{2 \cdot a}}} \]

   3. lift-/.f32 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

   4. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{g}{2}}{a}}} \]

   5. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

   7. lower-cbrt.f32 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{g}{2}}}}{\sqrt[3]{a}} \]

   9. lower-cbrt.f32 99.0

      \[\leadsto \frac{\sqrt[3]{\frac{g}{2}}}{\color{blue}{\sqrt[3]{a}}} \]

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{g}{2}}}{\sqrt[3]{a}}} \]

5. Taylor expanded in g around 0

   \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{1}{2} \cdot g}}}{\sqrt[3]{a}} \]
6. Step-by-step derivation

   1. lower-*.f32 99.0

      \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{g}}}{\sqrt[3]{a}} \]

7. Applied rewrites 99.0%

   \[\leadsto \frac{\sqrt[3]{\color{blue}{0.5 \cdot g}}}{\sqrt[3]{a}} \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g} \end{array} \]

FPCore

(FPCore (g a) :precision binary32 (* (cbrt (/ 0.5 a)) (cbrt g)))

C

float code(float g, float a) {
	return cbrtf((0.5f / a)) * cbrtf(g);
}

Julia

function code(g, a)
	return Float32(cbrt(Float32(Float32(0.5) / a)) * cbrt(g))
end

Derivation

1. Initial program 82.8%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cbrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{2 \cdot a}}} \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{2 \cdot a}}} \]

   3. lift-/.f32 N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{2 \cdot a}}} \]

   4. cbrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}} \]

   6. lower-cbrt.f32 N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{g}}}{\sqrt[3]{2 \cdot a}} \]

   7. lower-cbrt.f32 N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\color{blue}{\sqrt[3]{2 \cdot a}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a \cdot 2}}} \]

   9. lower-*.f32 98.5

      \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{\color{blue}{a \cdot 2}}} \]

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a \cdot 2}}} \]

5. Taylor expanded in g around 0

   \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}} \cdot \frac{1}{\sqrt[3]{2}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{1}{\sqrt[3]{2}} \]

   2. cbrt-div N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}}} \cdot \frac{1}{\sqrt[3]{2}} \]

   3. associate-/r* N/A

      \[\leadsto \sqrt[3]{\color{blue}{\frac{g}{a}}} \cdot \frac{1}{\sqrt[3]{2}} \]

   4. cbrt-undiv N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}}} \cdot \frac{1}{\sqrt[3]{2}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \color{blue}{\sqrt[3]{\frac{g}{a}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \color{blue}{\sqrt[3]{\frac{g}{a}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\color{blue}{\frac{g}{a}}} \]

   8. lower-cbrt.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\frac{g}{\color{blue}{a}}} \]

   9. lower-cbrt.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\frac{g}{a}} \]

   10. lift-/.f32 83.1

       \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\frac{g}{a}} \]

7. Applied rewrites 83.1%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\frac{g}{a}}} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \color{blue}{\sqrt[3]{\frac{g}{a}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\color{blue}{\frac{g}{a}}} \]

   3. lift-cbrt.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\frac{g}{\color{blue}{a}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\frac{g}{a}} \]

   5. lift-cbrt.f32 N/A

      \[\leadsto \frac{1}{\sqrt[3]{2}} \cdot \sqrt[3]{\frac{g}{a}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{\frac{1}{\sqrt[3]{2}}} \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{\frac{1}{\sqrt[3]{2}}} \]

   8. lift-cbrt.f32 N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{\color{blue}{1}}{\sqrt[3]{2}} \]

   9. lift-/.f32 N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{1}{\sqrt[3]{2}} \]

   10. lift-cbrt.f32 N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{1}{\sqrt[3]{2}} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\color{blue}{2}}} \]

   12. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{\sqrt[3]{-1 \cdot -1}}{\sqrt[3]{2}} \]

   13. cbrt-unprod N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{\sqrt[3]{\color{blue}{2}}} \]

   14. associate-*r/ N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{-1} \cdot \color{blue}{\frac{\sqrt[3]{-1}}{\sqrt[3]{2}}}\right) \]

   15. lift-cbrt.f32 N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{-1} \cdot \frac{\sqrt[3]{-1}}{\sqrt[3]{2}}\right) \]

   16. cbrt-undiv N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{\frac{-1}{2}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{\frac{-1}{2}}\right) \]

   18. cbrt-unprod N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{-1 \cdot \frac{-1}{2}} \]

   19. metadata-eval N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

   20. lower-cbrt.f32 83.1

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{0.5} \]

9. Applied rewrites 83.1%

   \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{\sqrt[3]{0.5}} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \color{blue}{\sqrt[3]{\frac{1}{2}}} \]

    2. lift-/.f32 N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{1}{2}} \]

    3. lift-cbrt.f32 N/A

       \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\color{blue}{\frac{1}{2}}} \]

    4. cbrt-div N/A

       \[\leadsto \frac{\sqrt[3]{g}}{\sqrt[3]{a}} \cdot \sqrt[3]{\color{blue}{\frac{1}{2}}} \]

    5. associate-*l/ N/A

       \[\leadsto \frac{\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{2}}}{\color{blue}{\sqrt[3]{a}}} \]

    6. associate-/l* N/A

       \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}} \]

    7. *-commutative N/A

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \color{blue}{\sqrt[3]{g}} \]

    8. lower-*.f32 N/A

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \color{blue}{\sqrt[3]{g}} \]

    9. lift-cbrt.f32 N/A

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \sqrt[3]{g} \]

    10. cbrt-undiv N/A

        \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\color{blue}{g}} \]

    11. lower-cbrt.f32 N/A

        \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{\color{blue}{g}} \]

    12. lower-/.f32 N/A

        \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot \sqrt[3]{g} \]

    13. lift-cbrt.f32 98.6

        \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{g} \]

11. Applied rewrites 98.6%

    \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \color{blue}{\sqrt[3]{g}} \]
12. Add Preprocessing

Alternative 3: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0.5 \cdot \frac{g}{a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary32 (cbrt (* 0.5 (/ g a))))

C

float code(float g, float a) {
	return cbrtf((0.5f * (g / a)));
}

Julia

function code(g, a)
	return cbrt(Float32(Float32(0.5) * Float32(g / a)))
end

Derivation

1. Initial program 82.8%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing

3. Taylor expanded in g around 0

   \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{2} \cdot \frac{g}{a}}} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt[3]{\frac{1}{2} \cdot \color{blue}{\frac{g}{a}}} \]

   2. lower-/.f32 83.2

      \[\leadsto \sqrt[3]{0.5 \cdot \frac{g}{\color{blue}{a}}} \]

5. Applied rewrites 83.2%

   \[\leadsto \sqrt[3]{\color{blue}{0.5 \cdot \frac{g}{a}}} \]
6. Add Preprocessing

Alternative 4: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g}{a + a}} \end{array} \]

FPCore

(FPCore (g a) :precision binary32 (cbrt (/ g (+ a a))))

C

float code(float g, float a) {
	return cbrtf((g / (a + a)));
}

Julia

function code(g, a)
	return cbrt(Float32(g / Float32(a + a)))
end

Derivation

1. Initial program 82.8%

   \[\sqrt[3]{\frac{g}{2 \cdot a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{2 \cdot a}}} \]

   2. count-2-rev N/A

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

   3. lower-+.f32 82.8

      \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]

4. Applied rewrites 82.8%

   \[\leadsto \sqrt[3]{\frac{g}{\color{blue}{a + a}}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025054 
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  :precision binary32
  (cbrt (/ g (* 2.0 a))))