Trowbridge-Reitz Sample, near normal, slope_x

Average Error: 0.3 → 0.3

Time: 14.2s

Precision: binary32

Cost: 6816

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

Math

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

↓

\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(-12.56637061436 \cdot \frac{0.5}{\frac{-1}{u2}}\right) \]

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* -12.56637061436 (/ 0.5 (/ -1.0 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 sqrtf((u1 / (1.0f - u1))) * cosf((-12.56637061436f * (0.5f / (-1.0f / 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 = sqrt((u1 / (1.0e0 - u1))) * cos(((-12.56637061436e0) * (0.5e0 / ((-1.0e0) / 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(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(-12.56637061436) * Float32(Float32(0.5) / Float32(Float32(-1.0) / 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 = sqrt((u1 / (single(1.0) - u1))) * cos((single(-12.56637061436) * (single(0.5) / (single(-1.0) / 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 0.3

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

3. Applied egg-rr 0.3

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(-12.56637061436 \cdot \frac{0.5}{\frac{-1}{u2}}\right)} \]

4. Final simplification 0.3

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(-12.56637061436 \cdot \frac{0.5}{\frac{-1}{u2}}\right) \]

Alternatives

Alternative 1
Error	1.9
Cost	10020

\[\begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999989867210388:\\ \;\;\;\;\sqrt{\left(u1 + 1\right) \cdot u1} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	6752

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

Alternative 3
Error	3.3
Cost	6692

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	6688

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

Alternative 5
Error	9.0
Cost	3360

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

Alternative 6
Error	6.2
Cost	3360

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

Alternative 7
Error	11.7
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023073 
(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))))