Trowbridge-Reitz Sample, near normal, slope_x

Average Error: 1.02% → 1.04%

Time: 11.2s

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

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

↓

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

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))) (+ 1.0 (+ -1.0 (cos (* u2 6.28318530718))))))

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))) * (1.0f + (-1.0f + cosf((u2 * 6.28318530718f))));
}

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))) * (1.0e0 + ((-1.0e0) + cos((u2 * 6.28318530718e0))))
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))) * Float32(Float32(1.0) + Float32(Float32(-1.0) + cos(Float32(u2 * Float32(6.28318530718))))))
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))) * (single(1.0) + (single(-1.0) + cos((u2 * single(6.28318530718)))));
end

Derivation

1. Initial program 1.02

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

2. Applied egg-rr 1.01

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

3. Simplified 1.01

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

   [Start] 1.01	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   *-commutative [=>] 1.01	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right) \]

4. Applied egg-rr 1.27

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(u2 \cdot 6.28318530718\right)\right)} - 1\right)} \]

5. Applied egg-rr 1.04

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

6. Final simplification 1.04

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

Alternatives

Alternative 1
Error	6.18%
Cost	9892

\[\begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9950000047683716:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 2
Error	1.02%
Cost	6688

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

Alternative 3
Error	12.25%
Cost	3616

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

Alternative 4
Error	28.9%
Cost	3360

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

Alternative 5
Error	20.62%
Cost	3360

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

Alternative 6
Error	37.06%
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023121 
(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))))