Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 14.2s

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

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

↓

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

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 (* u2 (* u2 39.47841760436263))))))

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((u2 * (u2 * 39.47841760436263f))));
}

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((u2 * (u2 * 39.47841760436263e0))))
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(u2 * Float32(u2 * Float32(39.47841760436263))))))
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((u2 * (u2 * single(39.47841760436263)))));
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

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

3. Simplified 0.5

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

   [Start] 0.5	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   *-commutative [=>] 0.5	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right) \]
   associate-*l* [=>] 0.5	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\color{blue}{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}}\right) \]

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	7072

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

Alternative 2
Error	2.0
Cost	6820

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\frac{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3
Error	3.1
Cost	6692

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004600000102072954:\\ \;\;\;\;\sqrt{\frac{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	6688

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

Alternative 5
Error	0.5
Cost	6688

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

Alternative 6
Error	5.8
Cost	3552

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

Alternative 7
Error	5.8
Cost	3552

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

Alternative 8
Error	5.9
Cost	3488

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

Alternative 9
Error	5.9
Cost	3488

\[6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1 - u1}{u1}}} \]

Alternative 10
Error	5.9
Cost	3488

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

Alternative 11
Error	5.8
Cost	3488

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

Alternative 12
Error	11.4
Cost	3424

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

Alternative 13
Error	11.4
Cost	3424

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

Alternative 14
Error	11.4
Cost	3360

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

Alternative 15
Error	25.4
Cost	288

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

Alternative 16
Error	25.4
Cost	224

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

Alternative 17
Error	25.4
Cost	224

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

Alternative 18
Error	25.8
Cost	160

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

Reproduce

herbie shell --seed 2023060 
(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))))