Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 45.0s

Precision: binary32

Cost: 10016

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

Math

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

↓

\[\left(\sqrt{\frac{u1}{0.5 + \frac{u1}{-2}}} \cdot \sqrt{0.5}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

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 (+ 0.5 (/ u1 -2.0)))) (sqrt 0.5))
  (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) {
	return (sqrtf((u1 / (0.5f + (u1 / -2.0f)))) * sqrtf(0.5f)) * 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
    code = (sqrt((u1 / (0.5e0 + (u1 / (-2.0e0))))) * sqrt(0.5e0)) * 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)
	return Float32(Float32(sqrt(Float32(u1 / Float32(Float32(0.5) + Float32(u1 / Float32(-2.0))))) * sqrt(Float32(0.5))) * 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)
	tmp = (sqrt((u1 / (single(0.5) + (u1 / single(-2.0))))) * sqrt(single(0.5))) * 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

   \[\leadsto \color{blue}{\left(\sqrt{\frac{u1}{0.5 + \frac{u1}{-2}}} \cdot \sqrt{0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Final simplification 0.5

   \[\leadsto \left(\sqrt{\frac{u1}{0.5 + \frac{u1}{-2}}} \cdot \sqrt{0.5}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

Alternatives

Alternative 1
Error	0.5
Cost	10016

\[\sqrt{0.5} \cdot \left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{0.5 + u1 \cdot -0.5}}\right) \]

Alternative 2
Error	0.5
Cost	10016

\[\left(\sqrt{0.5} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{0.5 + u1 \cdot -0.5}} \]

Alternative 3
Error	0.5
Cost	7072

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

Alternative 4
Error	0.5
Cost	6944

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

Alternative 5
Error	3.3
Cost	6692

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004220000002533197:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

Alternative 6
Error	0.5
Cost	6688

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

Alternative 7
Error	5.9
Cost	3488

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

Alternative 8
Error	6.0
Cost	3488

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

Alternative 9
Error	11.2
Cost	3424

\[\frac{u2 \cdot \left(\sqrt{u1} \cdot 12.56637061436\right)}{2} \]

Alternative 10
Error	11.2
Cost	3360

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

Alternative 11
Error	11.2
Cost	3360

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

Reproduce

herbie shell --seed 2023099 
(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))))