Trowbridge-Reitz Sample, near normal, slope_y

Average Accuracy: 98.3% → 98.3%

Time: 15.9s

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

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

↓

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

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 u1) -1.0))))

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 / u1) + -1.0f));
}

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 / u1) + (-1.0e0)))
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(Float32(1.0) / u1) + Float32(-1.0))))
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) / u1) + single(-1.0)));
end

Derivation

1. Initial program 98.3%

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

2. Applied egg-rr 98.4%

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

3. Applied egg-rr 98.3%

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

4. Simplified 98.2%

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

   [Start] 98.3	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1 - u1}{u1}}} \]
   div-sub [=>] 98.2	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]
   *-inverses [=>] 98.2	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1} - \color{blue}{1}}} \]
   sub-neg [=>] 98.2	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]
   metadata-eval [=>] 98.2	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

5. Applied egg-rr 98.3%

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

6. Simplified 98.3%

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

   [Start] 98.3	\[ \frac{\sin \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right)}{\sqrt{\frac{1}{u1} + -1}} \]
   associate-*l* [=>] 98.3	\[ \frac{\sin \left(\sqrt{\color{blue}{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

7. Final simplification 98.3%

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

Alternatives

Alternative 1
Accuracy	98.4%
Cost	9952

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

Alternative 2
Accuracy	98.3%
Cost	6880

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

Alternative 3
Accuracy	93.9%
Cost	6820

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{u1 \cdot \frac{u2 \cdot 39.47841760436263}{\frac{1 - u1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 4
Accuracy	90.2%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.01119999960064888:\\ \;\;\;\;\sqrt{u1 \cdot \frac{u2 \cdot 39.47841760436263}{\frac{1 - u1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \frac{1}{{u1}^{-0.5}}\\ \end{array} \]

Alternative 5
Accuracy	90.1%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.01899999938905239:\\ \;\;\;\;\sqrt{u1 \cdot \frac{u2 \cdot 39.47841760436263}{\frac{1 - u1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{u1}^{-0.5}}{\sin \left(u2 \cdot 6.28318530718\right)}}\\ \end{array} \]

Alternative 6
Accuracy	90.2%
Cost	6756

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.01119999960064888:\\ \;\;\;\;\sqrt{u1 \cdot \frac{u2 \cdot 39.47841760436263}{\frac{1 - u1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{\sin \left(u2 \cdot 6.28318530718\right)}}\\ \end{array} \]

Alternative 7
Accuracy	90.2%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.01119999960064888:\\ \;\;\;\;\sqrt{u1 \cdot \frac{u2 \cdot 39.47841760436263}{\frac{1 - u1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{{u1}^{-0.5}}\\ \end{array} \]

Alternative 8
Accuracy	90.2%
Cost	6692

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.01119999960064888:\\ \;\;\;\;\sqrt{u1 \cdot \frac{u2 \cdot 39.47841760436263}{\frac{1 - u1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 9
Accuracy	98.3%
Cost	6688

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

Alternative 10
Accuracy	98.2%
Cost	6688

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

Alternative 11
Accuracy	81.5%
Cost	3552

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

Alternative 12
Accuracy	81.6%
Cost	3488

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

Alternative 13
Accuracy	81.6%
Cost	3488

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

Alternative 14
Accuracy	64.7%
Cost	3424

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

Alternative 15
Accuracy	64.7%
Cost	3424

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

Alternative 16
Accuracy	64.7%
Cost	3424

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

Alternative 17
Accuracy	4.6%
Cost	3360

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

Alternative 18
Accuracy	64.7%
Cost	3360

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

Alternative 19
Accuracy	64.7%
Cost	3360

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

Alternative 20
Accuracy	-0.0%
Cost	3296

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

Reproduce

herbie shell --seed 2023129 
(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))))