Trowbridge-Reitz Sample, near normal, slope_x

Average Error: 0.3 → 0.3

Time: 12.3s

Precision: binary32

Cost: 13152

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

Math

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

↓

\[\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\cos \left(6.28318530718 \cdot u2\right)}^{3}} \]

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt (* (pow (/ u1 (- 1.0 u1)) 1.5) (pow (cos (* 6.28318530718 u2)) 3.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

↓

float code(float cosTheta_i, float u1, float u2) {
	return cbrtf((powf((u1 / (1.0f - u1)), 1.5f) * powf(cosf((6.28318530718f * u2)), 3.0f)));
}

Julia

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

↓

function code(cosTheta_i, u1, u2)
	return cbrt(Float32((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5)) * (cos(Float32(Float32(6.28318530718) * u2)) ^ Float32(3.0))))
end

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\cos \left(6.28318530718 \cdot u2\right)}^{3}}} \]

3. Final simplification 0.3

   \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\cos \left(6.28318530718 \cdot u2\right)}^{3}} \]

Alternatives

Alternative 1
Error	1.2
Cost	10020

\[\begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9950000047683716:\\ \;\;\;\;t_0 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	9920

\[\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}} \]

Alternative 3
Error	1.9
Cost	9892

\[\begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9900000095367432:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7072

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1 \cdot u1}\\ \cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{t_0 + u1 \cdot t_0} \end{array} \]

Alternative 5
Error	0.3
Cost	6688

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

Alternative 6
Error	3.8
Cost	3616

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

Alternative 7
Error	3.8
Cost	3616

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \]

Alternative 8
Error	5.4
Cost	3556

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.00047500000800937414:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 9
Error	9.1
Cost	3360

\[\sqrt{u1 \cdot \left(u1 + 1\right)} \]

Alternative 10
Error	6.5
Cost	3360

\[\sqrt{\frac{u1}{1 - u1}} \]

Alternative 11
Error	11.8
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2022354 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))