Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.6

Time: 11.7s

Precision: binary32

Cost: 9952

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

Math

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

↓

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

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 (sqrt (* u2 (* u2 39.47841760436263)))) (sqrt (+ -1.0 (/ 1.0 u1)))))

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(sqrtf((u2 * (u2 * 39.47841760436263f)))) / sqrtf((-1.0f + (1.0f / u1)));
}

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(sqrt((u2 * (u2 * 39.47841760436263e0)))) / sqrt(((-1.0e0) + (1.0e0 / u1)))
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(sqrt(Float32(u2 * Float32(u2 * Float32(39.47841760436263))))) / sqrt(Float32(Float32(-1.0) + Float32(Float32(1.0) / u1))))
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(sqrt((u2 * (u2 * single(39.47841760436263))))) / sqrt((single(-1.0) + (single(1.0) / u1)));
end

Derivation

1. Initial program 0.5

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

2. Applied egg-rr 0.6

   \[\leadsto \sqrt{\color{blue}{{\left({\left(\frac{1}{u1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Simplified 0.6

   \[\leadsto \sqrt{\color{blue}{\frac{1}{-1 + \frac{1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   Proof
   (/.f32 1 (+.f32 -1 (/.f32 1 u1))): 0 points increase in error, 0 points decrease in error
   (/.f32 1 (Rewrite<= +-commutative_binary32 (+.f32 (/.f32 1 u1) -1))): 0 points increase in error, 0 points decrease in error
   (/.f32 1 (+.f32 (/.f32 1 u1) (Rewrite<= metadata-eval (neg.f32 1)))): 0 points increase in error, 0 points decrease in error
   (/.f32 1 (Rewrite<= sub-neg_binary32 (-.f32 (/.f32 1 u1) 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unpow-1_binary32 (pow.f32 (-.f32 (/.f32 1 u1) 1) -1)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> sqr-pow_binary32 (*.f32 (pow.f32 (-.f32 (/.f32 1 u1) 1) (/.f32 -1 2)) (pow.f32 (-.f32 (/.f32 1 u1) 1) (/.f32 -1 2)))): 61 points increase in error, 71 points decrease in error
   (Rewrite<= fabs-sqr_binary32 (fabs.f32 (*.f32 (pow.f32 (-.f32 (/.f32 1 u1) 1) (/.f32 -1 2)) (pow.f32 (-.f32 (/.f32 1 u1) 1) (/.f32 -1 2))))): 0 points increase in error, 0 points decrease in error
   (fabs.f32 (Rewrite<= sqr-pow_binary32 (pow.f32 (-.f32 (/.f32 1 u1) 1) -1))): 71 points increase in error, 61 points decrease in error
   (fabs.f32 (Rewrite<= exp-to-pow_binary32 (exp.f32 (*.f32 (log.f32 (-.f32 (/.f32 1 u1) 1)) -1)))): 104 points increase in error, 105 points decrease in error
   (Rewrite<= rem-sqrt-square_binary32 (sqrt.f32 (*.f32 (exp.f32 (*.f32 (log.f32 (-.f32 (/.f32 1 u1) 1)) -1)) (exp.f32 (*.f32 (log.f32 (-.f32 (/.f32 1 u1) 1)) -1))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f32 (*.f32 (Rewrite=> exp-to-pow_binary32 (pow.f32 (-.f32 (/.f32 1 u1) 1) -1)) (exp.f32 (*.f32 (log.f32 (-.f32 (/.f32 1 u1) 1)) -1)))): 91 points increase in error, 95 points decrease in error
   (sqrt.f32 (*.f32 (pow.f32 (-.f32 (/.f32 1 u1) 1) -1) (Rewrite=> exp-to-pow_binary32 (pow.f32 (-.f32 (/.f32 1 u1) 1) -1)))): 86 points increase in error, 88 points decrease in error
   (sqrt.f32 (Rewrite=> pow-sqr_binary32 (pow.f32 (-.f32 (/.f32 1 u1) 1) (*.f32 2 -1)))): 20 points increase in error, 20 points decrease in error
   (sqrt.f32 (pow.f32 (-.f32 (/.f32 1 u1) 1) (Rewrite=> metadata-eval -2))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unpow1/2_binary32 (pow.f32 (pow.f32 (-.f32 (/.f32 1 u1) 1) -2) 1/2)): 0 points increase in error, 0 points decrease in error

4. Taylor expanded in u2 around inf 0.6

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

5. Simplified 0.6

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{-1 + \frac{1}{u1}}}} \]
   Proof
   (/.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (sqrt.f32 (+.f32 -1 (/.f32 1 u1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= rem-exp-log_binary32 (exp.f32 (log.f32 (/.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (sqrt.f32 (+.f32 -1 (/.f32 1 u1))))))): 210 points increase in error, 11 points decrease in error
   (exp.f32 (Rewrite=> log-div_binary32 (-.f32 (log.f32 (sin.f32 (*.f32 314159265359/50000000000 u2))) (log.f32 (sqrt.f32 (+.f32 -1 (/.f32 1 u1))))))): 45 points increase in error, 5 points decrease in error
   (exp.f32 (Rewrite=> sub-neg_binary32 (+.f32 (log.f32 (sin.f32 (*.f32 314159265359/50000000000 u2))) (neg.f32 (log.f32 (sqrt.f32 (+.f32 -1 (/.f32 1 u1)))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> exp-sum_binary32 (*.f32 (exp.f32 (log.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)))) (exp.f32 (neg.f32 (log.f32 (sqrt.f32 (+.f32 -1 (/.f32 1 u1)))))))): 47 points increase in error, 134 points decrease in error
   (*.f32 (Rewrite=> rem-exp-log_binary32 (sin.f32 (*.f32 314159265359/50000000000 u2))) (exp.f32 (neg.f32 (log.f32 (sqrt.f32 (+.f32 -1 (/.f32 1 u1))))))): 44 points increase in error, 126 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (sqrt.f32 (Rewrite<= remove-double-div_binary32 (/.f32 1 (/.f32 1 (+.f32 -1 (/.f32 1 u1)))))))))): 4 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (sqrt.f32 (Rewrite<= unpow-1_binary32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) -1))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (sqrt.f32 (Rewrite=> sqr-pow_binary32 (*.f32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (/.f32 -1 2)) (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (/.f32 -1 2))))))))): 3 points increase in error, 4 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (Rewrite=> rem-sqrt-square_binary32 (fabs.f32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (/.f32 -1 2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (fabs.f32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (Rewrite=> metadata-eval -1/2))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (fabs.f32 (Rewrite=> sqr-pow_binary32 (*.f32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (/.f32 -1/2 2)) (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (/.f32 -1/2 2))))))))): 24 points increase in error, 5 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (Rewrite=> fabs-sqr_binary32 (*.f32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (/.f32 -1/2 2)) (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (/.f32 -1/2 2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (Rewrite<= sqr-pow_binary32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) -1/2)))))): 5 points increase in error, 24 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (Rewrite<= metadata-eval (/.f32 -1 2))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (log.f32 (pow.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))) (Rewrite=> metadata-eval -1/2)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (Rewrite=> log-pow_binary32 (*.f32 -1/2 (log.f32 (/.f32 1 (+.f32 -1 (/.f32 1 u1))))))))): 3 points increase in error, 10 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (*.f32 -1/2 (log.f32 (Rewrite<= unpow-1_binary32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (*.f32 -1/2 (Rewrite=> log-pow_binary32 (*.f32 -1 (log.f32 (+.f32 -1 (/.f32 1 u1))))))))): 4 points increase in error, 4 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (neg.f32 (*.f32 -1/2 (Rewrite<= *-commutative_binary32 (*.f32 (log.f32 (+.f32 -1 (/.f32 1 u1))) -1)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (Rewrite<= distribute-rgt-neg-out_binary32 (*.f32 -1/2 (neg.f32 (*.f32 (log.f32 (+.f32 -1 (/.f32 1 u1))) -1)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (*.f32 -1/2 (Rewrite=> distribute-rgt-neg-in_binary32 (*.f32 (log.f32 (+.f32 -1 (/.f32 1 u1))) (neg.f32 -1)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (*.f32 -1/2 (*.f32 (log.f32 (+.f32 -1 (/.f32 1 u1))) (Rewrite=> metadata-eval 1))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (*.f32 -1/2 (Rewrite=> *-rgt-identity_binary32 (log.f32 (+.f32 -1 (/.f32 1 u1))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (exp.f32 (Rewrite=> *-commutative_binary32 (*.f32 (log.f32 (+.f32 -1 (/.f32 1 u1))) -1/2)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (Rewrite=> exp-to-pow_binary32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2))): 26 points increase in error, 125 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (Rewrite<= unpow1_binary32 (pow.f32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2) 1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (Rewrite=> sqr-pow_binary32 (*.f32 (pow.f32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2) (/.f32 1 2)) (pow.f32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2) (/.f32 1 2))))): 76 points increase in error, 21 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (Rewrite<= fabs-sqr_binary32 (fabs.f32 (*.f32 (pow.f32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2) (/.f32 1 2)) (pow.f32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2) (/.f32 1 2)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (fabs.f32 (Rewrite<= sqr-pow_binary32 (pow.f32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2) 1)))): 21 points increase in error, 76 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (fabs.f32 (Rewrite=> unpow1_binary32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (Rewrite<= rem-sqrt-square_binary32 (sqrt.f32 (*.f32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2) (pow.f32 (+.f32 -1 (/.f32 1 u1)) -1/2))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (sqrt.f32 (Rewrite=> pow-sqr_binary32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) (*.f32 2 -1/2))))): 8 points increase in error, 13 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (sqrt.f32 (pow.f32 (+.f32 -1 (/.f32 1 u1)) (Rewrite=> metadata-eval -1)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (sqrt.f32 (Rewrite=> unpow-1_binary32 (/.f32 1 (+.f32 -1 (/.f32 1 u1)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (sqrt.f32 (/.f32 1 (Rewrite=> +-commutative_binary32 (+.f32 (/.f32 1 u1) -1))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (sqrt.f32 (/.f32 1 (+.f32 (/.f32 1 u1) (Rewrite<= metadata-eval (neg.f32 1)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sin.f32 (*.f32 314159265359/50000000000 u2)) (sqrt.f32 (/.f32 1 (Rewrite<= sub-neg_binary32 (-.f32 (/.f32 1 u1) 1))))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.5

   \[\leadsto \frac{\sin \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)}}{\sqrt{-1 + \frac{1}{u1}}} \]

7. Simplified 0.6

   \[\leadsto \frac{\sin \color{blue}{\left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right)}}{\sqrt{-1 + \frac{1}{u1}}} \]
   Proof
   (sqrt.f32 (*.f32 u2 (*.f32 u2 98696044010906577398881/2500000000000000000000))): 0 points increase in error, 0 points decrease in error
   (sqrt.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 u2 u2) 98696044010906577398881/2500000000000000000000))): 23 points increase in error, 24 points decrease in error
   (sqrt.f32 (Rewrite<= *-commutative_binary32 (*.f32 98696044010906577398881/2500000000000000000000 (*.f32 u2 u2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unpow1/2_binary32 (pow.f32 (*.f32 98696044010906577398881/2500000000000000000000 (*.f32 u2 u2)) 1/2)): 0 points increase in error, 0 points decrease in error

8. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.5
Cost	9952

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

Alternative 2
Error	3.2
Cost	6692

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004999999888241291:\\ \;\;\;\;\sqrt{u2 \cdot \frac{u2}{\frac{0.02533029591058111}{u1} + -0.02533029591058111}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	6688

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

Alternative 4
Error	0.6
Cost	6688

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

Alternative 5
Error	6.1
Cost	3488

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

Alternative 6
Error	11.5
Cost	3360

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

Alternative 7
Error	11.5
Cost	3360

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

Alternative 8
Error	25.5
Cost	224

\[6.28318530718 \cdot \left(u2 \cdot \left(u1 + 0.5\right)\right) \]

Alternative 9
Error	25.8
Cost	160

\[6.28318530718 \cdot \left(u2 \cdot u1\right) \]

Alternative 10
Error	25.8
Cost	160

\[u2 \cdot \left(u1 \cdot 6.28318530718\right) \]

Reproduce

herbie shell --seed 2022329 
(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))))