Trowbridge-Reitz Sample, near normal, slope_x

Average Error: 99.0% → 99.0%

Time: 23.7s

Precision: binary32

Cost: 7072.00

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

Math

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

↓

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

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 (* u1 u1)))))
   (* (sqrt (+ t_0 (* u1 t_0))) (cos (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

↓

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - (u1 * u1));
	return sqrtf((t_0 + (u1 * t_0))) * cosf((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))) * cos((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 = u1 / (1.0e0 - (u1 * u1))
    code = sqrt((t_0 + (u1 * t_0))) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(6.28318530718) * u2)))
end

↓

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - Float32(u1 * u1)))
	return Float32(sqrt(Float32(t_0 + Float32(u1 * t_0))) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((single(6.28318530718) * u2));
end

↓

function tmp = code(cosTheta_i, u1, u2)
	t_0 = u1 / (single(1.0) - (u1 * u1));
	tmp = sqrt((t_0 + (u1 * t_0))) * cos((single(6.28318530718) * u2));
end

Derivation

1. Initial program 99.0

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

2. Applied egg-rr 99.0

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

3. Final simplification 99.0

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

Alternatives

Alternative 1
Error	96.4%
Cost	6820.00

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]

Alternative 2
Error	94.3%
Cost	6692.00

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.2199999988079071:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3
Error	99.0%
Cost	6688.00

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

Alternative 4
Error	88.8%
Cost	3616.00

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

Alternative 5
Error	88.8%
Cost	3616.00

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \]

Alternative 6
Error	80.3%
Cost	3392.00

\[{\left(\frac{u1}{1 - u1}\right)}^{0.5} \]

Alternative 7
Error	72.2%
Cost	3360.00

\[\sqrt{u1 \cdot \left(u1 + 1\right)} \]

Alternative 8
Error	72.2%
Cost	3360.00

\[\sqrt{u1 + u1 \cdot u1} \]

Alternative 9
Error	80.3%
Cost	3360.00

\[\sqrt{\frac{u1}{1 - u1}} \]

Alternative 10
Error	63.9%
Cost	3232.00

\[\sqrt{u1} \]

Alternative 11
Error	20.1%
Cost	96.00

\[u1 + 0.5 \]

Alternative 12
Error	19.1%
Cost	32.00

\[u1 \]

Reproduce

herbie shell --seed 2023093 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))