Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 13.9s

Precision: binary32

Cost: 6720

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

Math

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

↓

\[\sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5} \]

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* 6.28318530718 u2)) (pow (/ u1 (- 1.0 u1)) 0.5)))

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 sinf((6.28318530718f * u2)) * powf((u1 / (1.0f - u1)), 0.5f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

↓

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin((6.28318530718e0 * u2)) * ((u1 / (1.0e0 - u1)) ** 0.5e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

↓

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(6.28318530718) * u2)) * (Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

↓

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(6.28318530718) * u2)) * ((u1 / (single(1.0) - u1)) ^ single(0.5));
end

Derivation

1. Initial program 0.5

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

2. Applied egg-rr 0.9

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

3. Applied egg-rr 0.7

   \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \left({\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

4. Simplified 0.5

   \[\leadsto \color{blue}{\sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}} \]
   Proof
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/2)): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (pow.f32 (/.f32 u1 (-.f32 1 u1)) (Rewrite<= metadata-eval (*.f32 2 1/4)))): 2 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (Rewrite<= pow-sqr_binary32 (*.f32 (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/4) (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/4)))): 0 points increase in error, 3 points decrease in error
   (Rewrite=> *-commutative_binary32 (*.f32 (*.f32 (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/4) (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/4)) (sin.f32 (*.f32 314159265359/50000000000 u2)))): 0 points increase in error, 3 points decrease in error
   (Rewrite<= associate-*r*_binary32 (*.f32 (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/4) (*.f32 (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/4) (sin.f32 (*.f32 314159265359/50000000000 u2))))): 0 points increase in error, 2 points decrease in error

5. Final simplification 0.5

   \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5} \]

Alternatives

Alternative 1
Error	1.9
Cost	6820

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.002400000113993883:\\ \;\;\;\;\sqrt{\left(u1 \cdot \frac{u2 \cdot u2}{1 - u1}\right) \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 2
Error	3.0
Cost	6756

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009050000458955765:\\ \;\;\;\;\sqrt{\left(u1 \cdot \frac{u2 \cdot u2}{1 - u1}\right) \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{\sin \left(6.28318530718 \cdot u2\right)}}\\ \end{array} \]

Alternative 3
Error	3.0
Cost	6692

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009050000458955765:\\ \;\;\;\;\sqrt{\left(u1 \cdot \frac{u2 \cdot u2}{1 - u1}\right) \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	6688

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

Alternative 5
Error	4.3
Cost	4004

\[\begin{array}{l} t_0 := u1 \cdot 0.5 + -1\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.002400000113993883:\\ \;\;\;\;\sqrt{\left(u1 \cdot \frac{u2 \cdot u2}{1 - u1}\right) \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{-1.0471975511966667 \cdot \left(u2 \cdot t_0\right) + -0.15915494309188485 \cdot \frac{t_0}{u2}}\\ \end{array} \]

Alternative 6
Error	5.8
Cost	3552

\[\sqrt{\left(u1 \cdot \frac{u2 \cdot u2}{1 - u1}\right) \cdot 39.47841760436263} \]

Alternative 7
Error	5.8
Cost	3488

\[\sqrt{u2 \cdot \frac{u2}{\frac{0.02533029591058111}{u1} + -0.02533029591058111}} \]

Alternative 8
Error	5.8
Cost	3488

\[u2 \cdot \sqrt{\frac{39.47841760436263}{-1 + \frac{1}{u1}}} \]

Alternative 9
Error	11.3
Cost	3360

\[6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]

Alternative 10
Error	11.3
Cost	3360

\[u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right) \]

Alternative 11
Error	32.0
Cost	3296

\[u2 \cdot \sqrt{-39.47841760436263} \]

Reproduce

herbie shell --seed 2022343 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))