Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 14.4s

Precision: binary32

Cost: 10144

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

Math

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

↓

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

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (/ u1 (- 1.0 (* u1 u1))) (+ u1 1.0)))
  (sin (sqrt (* 39.47841760436263 (* u2 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) {
	return sqrtf(((u1 / (1.0f - (u1 * u1))) * (u1 + 1.0f))) * sinf(sqrtf((39.47841760436263f * (u2 * 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
    code = sqrt(((u1 / (1.0e0 - (u1 * u1))) * (u1 + 1.0e0))) * sin(sqrt((39.47841760436263e0 * (u2 * 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)
	return Float32(sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - Float32(u1 * u1))) * Float32(u1 + Float32(1.0)))) * sin(sqrt(Float32(Float32(39.47841760436263) * Float32(u2 * 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)
	tmp = sqrt(((u1 / (single(1.0) - (u1 * u1))) * (u1 + single(1.0)))) * sin(sqrt((single(39.47841760436263) * (u2 * 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

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

3. Applied egg-rr 0.5

   \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	9952

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

Alternative 2
Error	3.1
Cost	6692

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

Alternative 3
Error	0.5
Cost	6688

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

Alternative 4
Error	5.9
Cost	3552

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

Alternative 5
Error	5.9
Cost	3552

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

Alternative 6
Error	5.9
Cost	3552

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

Alternative 7
Error	6.0
Cost	3488

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

Alternative 8
Error	11.3
Cost	3424

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

Alternative 9
Error	11.3
Cost	3360

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

Alternative 10
Error	11.3
Cost	3360

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

Alternative 11
Error	25.8
Cost	160

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

Reproduce

herbie shell --seed 2022310 
(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))))