Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.6

Time: 13.4s

Precision: binary32

Cost: 10112

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

Math

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

↓

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

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 (* (/ 1.0 (+ (pow u1 -2.0) -1.0)) (+ 1.0 (/ 1.0 u1))))
  (sin (* 6.28318530718 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(((1.0f / (powf(u1, -2.0f) + -1.0f)) * (1.0f + (1.0f / u1)))) * sinf((6.28318530718f * 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(((1.0e0 / ((u1 ** (-2.0e0)) + (-1.0e0))) * (1.0e0 + (1.0e0 / u1)))) * sin((6.28318530718e0 * 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(Float32(Float32(1.0) / Float32((u1 ^ Float32(-2.0)) + Float32(-1.0))) * Float32(Float32(1.0) + Float32(Float32(1.0) / u1)))) * sin(Float32(Float32(6.28318530718) * 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(((single(1.0) / ((u1 ^ single(-2.0)) + single(-1.0))) * (single(1.0) + (single(1.0) / u1)))) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 0.5

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

2. Applied egg-rr 1.2

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

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.6

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

5. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	6880

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

Alternative 2
Error	0.6
Cost	6816

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

Alternative 3
Error	0.5
Cost	6752

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

Alternative 4
Error	3.1
Cost	6692

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.010499999858438969:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5
Error	0.6
Cost	6688

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

Alternative 6
Error	0.5
Cost	6688

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

Alternative 7
Error	5.9
Cost	3520

\[6.28318530718 \cdot \left(u2 \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5}\right) \]

Alternative 8
Error	5.9
Cost	3488

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

Alternative 9
Error	5.9
Cost	3488

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

Alternative 10
Error	11.2
Cost	3360

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

Alternative 11
Error	11.2
Cost	3360

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

Alternative 12
Error	25.5
Cost	288

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

Alternative 13
Error	25.5
Cost	224

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

Alternative 14
Error	30.5
Cost	160

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

Alternative 15
Error	25.8
Cost	160

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

Alternative 16
Error	25.8
Cost	160

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

Reproduce

herbie shell --seed 2022313 
(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))))