Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 13.9s

Precision: binary32

Cost: 7072

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

Math

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

↓

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

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u1 u1))))
   (* (sqrt (+ (/ (* u1 u1) t_0) (/ u1 t_0))) (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) {
	float t_0 = 1.0f - (u1 * u1);
	return sqrtf((((u1 * u1) / t_0) + (u1 / t_0))) * 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
    real(4) :: t_0
    t_0 = 1.0e0 - (u1 * u1)
    code = sqrt((((u1 * u1) / t_0) + (u1 / t_0))) * 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)
	t_0 = Float32(Float32(1.0) - Float32(u1 * u1))
	return Float32(sqrt(Float32(Float32(Float32(u1 * u1) / t_0) + Float32(u1 / t_0))) * 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)
	t_0 = single(1.0) - (u1 * u1);
	tmp = sqrt((((u1 * u1) / t_0) + (u1 / t_0))) * 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 0.5

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

3. Applied egg-rr 0.5

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

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	3.1
Cost	6692

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1 \cdot 39.47841760436263}{1 - u1} \cdot \left(u2 \cdot u2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	6688

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

Alternative 3
Error	5.8
Cost	3552

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

Alternative 4
Error	5.8
Cost	3552

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

Alternative 5
Error	5.9
Cost	3488

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

Alternative 6
Error	11.3
Cost	3424

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

Alternative 7
Error	25.7
Cost	3360

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

Alternative 8
Error	11.3
Cost	3360

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

Alternative 9
Error	25.8
Cost	160

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

Reproduce

herbie shell --seed 2022311 
(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))))