Trowbridge-Reitz Sample, near normal, slope_x

Average Error: 0.3 → 0.4

Time: 11.8s

Precision: binary32

Cost: 6720

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

Math

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

↓

\[{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (+ (/ 1.0 u1) -1.0) -0.5) (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) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * 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
    code = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * 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)
	return Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * 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)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * cos((single(6.28318530718) * u2));
end

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.4

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

3. Applied egg-rr 0.4

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right)} \]

4. Final simplification 0.4

   \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

Alternatives

Alternative 1
Error	1.2
Cost	6820

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]

Alternative 2
Error	1.2
Cost	6820

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3
Error	1.9
Cost	6692

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.30000001192092896:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	6688

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

Alternative 5
Error	3.8
Cost	3648

\[{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \]

Alternative 6
Error	3.8
Cost	3616

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

Alternative 7
Error	3.8
Cost	3616

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

Alternative 8
Error	5.3
Cost	3556

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.0010999999940395355:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 9
Error	9.1
Cost	3360

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

Alternative 10
Error	6.4
Cost	3360

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

Alternative 11
Error	11.7
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023054 
(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))))