Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 14.3s

Precision: binary32

Cost: 7712

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ t_1 := \frac{1}{t_0}\\ \sqrt{t_0 \cdot \left(t_0 \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{1}{t_1}\right)\right)\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

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 (/ u1 (- 1.0 u1))) (t_1 (/ 1.0 t_0)))
   (*
    (sqrt (* t_0 (* t_0 (* t_1 (* t_1 (/ 1.0 t_1))))))
    (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 = u1 / (1.0f - u1);
	float t_1 = 1.0f / t_0;
	return sqrtf((t_0 * (t_0 * (t_1 * (t_1 * (1.0f / t_1)))))) * 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
    real(4) :: t_1
    t_0 = u1 / (1.0e0 - u1)
    t_1 = 1.0e0 / t_0
    code = sqrt((t_0 * (t_0 * (t_1 * (t_1 * (1.0e0 / t_1)))))) * 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(u1 / Float32(Float32(1.0) - u1))
	t_1 = Float32(Float32(1.0) / t_0)
	return Float32(sqrt(Float32(t_0 * Float32(t_0 * Float32(t_1 * Float32(t_1 * Float32(Float32(1.0) / t_1)))))) * 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 = u1 / (single(1.0) - u1);
	t_1 = single(1.0) / t_0;
	tmp = sqrt((t_0 * (t_0 * (t_1 * (t_1 * (single(1.0) / t_1)))))) * 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

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

3. Applied egg-rr 0.5

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

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	7136

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

Alternative 2
Error	3.3
Cost	6692

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0024999999441206455:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{t_0 \cdot \left(t_0 \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{1}{t_1}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	6688

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

Alternative 4
Error	5.8
Cost	4512

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ t_1 := \frac{1}{t_0}\\ 6.28318530718 \cdot \left(u2 \cdot \sqrt{t_0 \cdot \left(t_0 \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{1}{t_1}\right)\right)\right)}\right) \end{array} \]

Alternative 5
Error	5.8
Cost	3936

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

Alternative 6
Error	5.8
Cost	3488

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

Alternative 7
Error	5.8
Cost	3488

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

Alternative 8
Error	11.3
Cost	3360

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

Reproduce

herbie shell --seed 2023090 
(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))))