Trowbridge-Reitz Sample, near normal, slope_y

Average Accuracy: 98.3% → 98.3%

Time: 15.0s

Precision: binary32

Cost: 6688

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

Math

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

↓

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

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 (* 6.28318530718 u2)) (sqrt (/ (- 1.0 u1) 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((6.28318530718f * u2)) / sqrtf(((1.0f - u1) / 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((6.28318530718e0 * u2)) / sqrt(((1.0e0 - u1) / 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(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(Float32(1.0) - u1) / 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((single(6.28318530718) * u2)) / sqrt(((single(1.0) - u1) / u1));
end

Derivation

1. Initial program 98.3%

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

2. Applied egg-rr 98.3%

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

   [Start] 98.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   clear-num [=>] 98.3	\[ \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   sqrt-div [=>] 98.1	\[ \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   metadata-eval [=>] 98.1	\[ \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   associate-*l/ [=>] 98.3	\[ \color{blue}{\frac{1 \cdot \sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]
   *-un-lft-identity [<=] 98.3	\[ \frac{\color{blue}{\sin \left(6.28318530718 \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

3. Final simplification 98.3%

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

Alternatives

Alternative 1
Accuracy	94.2%
Cost	6820

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005499999970197678:\\ \;\;\;\;\sqrt{\frac{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]

Alternative 2
Accuracy	90.4%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{\frac{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{u1}^{-0.5}}{\sin \left(6.28318530718 \cdot u2\right)}}\\ \end{array} \]

Alternative 3
Accuracy	90.5%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{\frac{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{{u1}^{-0.5}}\\ \end{array} \]

Alternative 4
Accuracy	90.5%
Cost	6692

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{\frac{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5
Accuracy	98.3%
Cost	6688

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

Alternative 6
Accuracy	98.2%
Cost	6688

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

Alternative 7
Accuracy	81.4%
Cost	3552

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

Alternative 8
Accuracy	81.4%
Cost	3552

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

Alternative 9
Accuracy	81.5%
Cost	3552

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

Alternative 10
Accuracy	81.5%
Cost	3552

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

Alternative 11
Accuracy	81.1%
Cost	3488

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

Alternative 12
Accuracy	81.3%
Cost	3488

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

Alternative 13
Accuracy	64.6%
Cost	3424

\[\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1}}} \]

Alternative 14
Accuracy	64.6%
Cost	3360

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

Alternative 15
Accuracy	-0.0%
Cost	3296

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

Reproduce

herbie shell --seed 2023138 
(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))))