Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 13.0s

Precision: binary32

\[cosTheta_i > 0.9999 \land cosTheta_i \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\]

Math

\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\]

↓

\[\sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)} \cdot \sin \left(\sqrt[3]{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right)\right) \cdot \left(6.28318530718 \cdot \left(u2 \cdot \left(u2 \cdot u2\right)\right)\right)}\right)\]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (/ u1 (- 1.0 (* u1 u1))) (+ u1 1.0)))
  (sin
   (cbrt
    (*
     (*
      (* (cbrt 6.28318530718) (cbrt 6.28318530718))
      (*
       (* (cbrt 6.28318530718) (cbrt 6.28318530718))
       (* (cbrt 6.28318530718) (cbrt 6.28318530718))))
     (* 6.28318530718 (* u2 (* u2 u2))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1 / (1.0f - u1)) * sinf(6.28318530718f * u2);
}

↓

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - (u1 * u1))) * (u1 + 1.0f)) * sinf(cbrtf(((cbrtf(6.28318530718f) * cbrtf(6.28318530718f)) * ((cbrtf(6.28318530718f) * cbrtf(6.28318530718f)) * (cbrtf(6.28318530718f) * cbrtf(6.28318530718f)))) * (6.28318530718f * (u2 * (u2 * u2)))));
}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Derivation

1. Initial program 0.5

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt_binary32 0.5

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \sqrt[3]{6.28318530718}\right)} \cdot u2\right)\]

4. Applied associate-*l*_binary32 0.5

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot u2\right)\right)}\]

5. Simplified 0.5

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \color{blue}{\left(u2 \cdot \sqrt[3]{6.28318530718}\right)}\right)\]
6. Using strategy rm

7. Applied flip--_binary32 0.5

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}}}} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(u2 \cdot \sqrt[3]{6.28318530718}\right)\right)\]

8. Applied associate-/r/_binary32 0.5

   \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot \left(1 + u1\right)}} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(u2 \cdot \sqrt[3]{6.28318530718}\right)\right)\]

9. Simplified 0.5

   \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1 \cdot u1}} \cdot \left(1 + u1\right)} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(u2 \cdot \sqrt[3]{6.28318530718}\right)\right)\]
10. Using strategy rm

11. Applied add-cbrt-cube_binary32 0.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(1 + u1\right)} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\color{blue}{\sqrt[3]{\left(u2 \cdot u2\right) \cdot u2}} \cdot \sqrt[3]{6.28318530718}\right)\right)\]

12. Applied cbrt-unprod_binary32 0.6

    \[\leadsto \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(1 + u1\right)} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \color{blue}{\sqrt[3]{\left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot 6.28318530718}}\right)\]

13. Applied add-cbrt-cube_binary32 0.6

    \[\leadsto \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(1 + u1\right)} \cdot \sin \left(\color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)}} \cdot \sqrt[3]{\left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot 6.28318530718}\right)\]

14. Applied cbrt-unprod_binary32 0.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(1 + u1\right)} \cdot \sin \color{blue}{\left(\sqrt[3]{\left(\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right) \cdot \left(\left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot 6.28318530718\right)}\right)}\]

15. Final simplification 0.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)} \cdot \sin \left(\sqrt[3]{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right)\right) \cdot \left(6.28318530718 \cdot \left(u2 \cdot \left(u2 \cdot u2\right)\right)\right)}\right)\]

Reproduce

herbie shell --seed 2021174 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0) (<= 2.328306437e-10 u1 1.0) (<= 2.328306437e-10 u2 1.0))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))