Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 19.5s

Precision: binary32

Cost: 6752

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

Math

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

↓

\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 - \frac{u2}{-0.18927975110791048}\right) \]

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 (- u2 (/ u2 -0.18927975110791048)))))

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((u2 - (u2 / -0.18927975110791048f)));
}

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((u2 - (u2 / (-0.18927975110791048e0))))
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(Float32(u2 - Float32(u2 / Float32(-0.18927975110791048)))))
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((u2 - (u2 / single(-0.18927975110791048))));
end

Derivation

1. Initial program 0.5

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

2. Applied egg-rr 0.5

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

3. Simplified 0.6

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

   [Start] 0.5	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2 - \left(u2 - u2\right)\right) \]
   rational.json-simplify-6 [=>] 0.6	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(u2 - \left(u2 - 6.28318530718 \cdot u2\right)\right)} \]

4. Taylor expanded in u2 around 0 0.6

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 - \color{blue}{-5.28318530718 \cdot u2}\right) \]

5. Simplified 0.6

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 - \color{blue}{u2 \cdot -5.28318530718}\right) \]
   Proof

   [Start] 0.6	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 - -5.28318530718 \cdot u2\right) \]
   rational.json-simplify-50 [=>] 0.6	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 - \color{blue}{u2 \cdot -5.28318530718}\right) \]

6. Applied egg-rr 0.5

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 - \color{blue}{\frac{u2}{-0.18927975110791048}}\right) \]

7. Final simplification 0.5

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 - \frac{u2}{-0.18927975110791048}\right) \]

Alternatives

Alternative 1
Error	2.1
Cost	13224

\[\begin{array}{l} t_0 := \sin \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq -0.05000000074505806:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 - \frac{u2}{-0.18927975110791048}\right)\\ \mathbf{elif}\;t_0 \leq 0.13500000536441803:\\ \;\;\;\;\frac{0.5}{\frac{u2 \cdot 0.5235987755983333 + \frac{0.07957747154594243}{u2}}{\sqrt{\frac{u1}{1 - u1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot t_0\\ \end{array} \]

Alternative 2
Error	2.1
Cost	13224

\[\begin{array}{l} t_0 := \sin \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq -0.05000000074505806:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\frac{6.28318530718}{\frac{1}{u2}}\right)\\ \mathbf{elif}\;t_0 \leq 0.13500000536441803:\\ \;\;\;\;\frac{0.5}{\frac{u2 \cdot 0.5235987755983333 + \frac{0.07957747154594243}{u2}}{\sqrt{\frac{u1}{1 - u1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot t_0\\ \end{array} \]

Alternative 3
Error	2.1
Cost	6692

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.13500000536441803:\\ \;\;\;\;\frac{0.5}{\frac{u2 \cdot 0.5235987755983333 + \frac{0.07957747154594243}{u2}}{\sqrt{\frac{u1}{1 - u1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	6688

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

Alternative 5
Error	3.7
Cost	3680

\[\sqrt{\frac{u1}{1 - u1}} \cdot \frac{1}{u2 \cdot 1.0471975511966667 + \frac{0.15915494309188485}{u2}} \]

Alternative 6
Error	3.7
Cost	3680

\[\frac{0.5}{\frac{u2 \cdot 0.5235987755983333 + \frac{0.07957747154594243}{u2}}{\sqrt{\frac{u1}{1 - u1}}}} \]

Alternative 7
Error	5.8
Cost	3488

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

Alternative 8
Error	5.8
Cost	3488

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

Alternative 9
Error	5.8
Cost	3488

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

Alternative 10
Error	11.3
Cost	3360

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

Alternative 11
Error	11.3
Cost	3360

\[u2 \cdot \frac{\sqrt{u1}}{0.15915494309188485} \]

Alternative 12
Error	11.3
Cost	3360

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

Alternative 13
Error	29.7
Cost	160

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

Reproduce

herbie shell --seed 2023073 
(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))))