Trowbridge-Reitz Sample, near normal, slope_y

Average Accuracy: 98.3% → 98.4%

Time: 16.5s

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

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

↓

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

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))) (sin (sqrt (* 39.47841760436263 (* 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))) * sinf(sqrtf((39.47841760436263f * (u2 * u2))));
}

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

Derivation

1. Initial program 98.1%

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

2. Applied egg-rr 98.4%

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

   [Start] 98.1	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   add-sqr-sqrt [=>] 97.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{6.28318530718 \cdot u2} \cdot \sqrt{6.28318530718 \cdot u2}\right)} \]
   pow1/2 [=>] 97.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}} \cdot \sqrt{6.28318530718 \cdot u2}\right) \]
   pow1/2 [=>] 97.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left({\left(6.28318530718 \cdot u2\right)}^{0.5} \cdot \color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}}\right) \]
   pow-prod-down [=>] 98.1	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left({\left(\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)\right)}^{0.5}\right)} \]
   swap-sqr [=>] 97.8	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left({\color{blue}{\left(\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)\right)}}^{0.5}\right) \]
   metadata-eval [=>] 98.4	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Simplified 98.4%

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

   [Start] 98.4	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]
   unpow1/2 [=>] 98.4	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

4. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	94.2%
Cost	16620

\[\begin{array}{l} t_0 := \sin \left(u2 \cdot 6.28318530718\right)\\ t_1 := \frac{u1}{1 - u1}\\ t_2 := \frac{\sqrt{u1}}{\frac{1}{t_0}}\\ \mathbf{if}\;t_0 \leq -0.05000000074505806:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{u2 \cdot \left(u2 \cdot \left(t_1 \cdot 39.47841760436263\right)\right)}\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\sqrt{t_1} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	94.2%
Cost	16556

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ t_1 := \sin \left(u2 \cdot 6.28318530718\right)\\ t_2 := t_1 \cdot \sqrt{u1}\\ \mathbf{if}\;t_1 \leq -0.05000000074505806:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{u2 \cdot \left(u2 \cdot \left(t_0 \cdot 39.47841760436263\right)\right)}\\ \mathbf{elif}\;t_1 \leq 0.20000000298023224:\\ \;\;\;\;\sqrt{t_0} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	98.3%
Cost	6688

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

Alternative 4
Accuracy	89.2%
Cost	3680

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \]

Alternative 5
Accuracy	81.8%
Cost	3552

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

Alternative 6
Accuracy	81.8%
Cost	3552

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

Alternative 7
Accuracy	81.4%
Cost	3488

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

Alternative 8
Accuracy	81.4%
Cost	3488

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

Alternative 9
Accuracy	81.7%
Cost	3488

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

Alternative 10
Accuracy	64.5%
Cost	3424

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

Alternative 11
Accuracy	64.5%
Cost	3424

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

Alternative 12
Accuracy	64.5%
Cost	3424

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

Alternative 13
Accuracy	64.5%
Cost	3360

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

Alternative 14
Accuracy	64.5%
Cost	3360

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

Alternative 15
Accuracy	20.6%
Cost	224

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

Alternative 16
Accuracy	20.6%
Cost	224

\[u2 \cdot \left(u1 \cdot 6.28318530718 + 3.14159265359\right) \]

Alternative 17
Accuracy	19.4%
Cost	160

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

Alternative 18
Accuracy	19.4%
Cost	160

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

Reproduce

herbie shell --seed 2023159 
(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))))