Trowbridge-Reitz Sample, near normal, slope_x

Average Error: 0.3 → 0.4

Time: 12.8s

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

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

↓

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

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 (/ 1.0 (+ (/ 1.0 u1) -1.0)))
  (cos (sqrt (* u2 (* u2 39.47841760436263))))))

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((1.0f / ((1.0f / u1) + -1.0f))) * cosf(sqrtf((u2 * (u2 * 39.47841760436263f))));
}

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((1.0e0 / ((1.0e0 / u1) + (-1.0e0)))) * cos(sqrt((u2 * (u2 * 39.47841760436263e0))))
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(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))) * cos(sqrt(Float32(u2 * Float32(u2 * Float32(39.47841760436263))))))
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((single(1.0) / ((single(1.0) / u1) + single(-1.0)))) * cos(sqrt((u2 * (u2 * single(39.47841760436263)))));
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

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

3. Simplified 0.3

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

   [Start] 0.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   *-commutative [=>] 0.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right) \]
   associate-*l* [=>] 0.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}}\right) \]

4. Applied egg-rr 0.3

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

5. Simplified 0.4

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

   [Start] 0.3	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 - u1 \cdot u1} \cdot 1} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]
   distribute-lft-in [<=] 0.4	\[ \sqrt{\color{blue}{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

6. Applied egg-rr 0.4

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

7. Taylor expanded in u1 around 0 0.4

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

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.5
Cost	9956

\[\begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.8700000047683716:\\ \;\;\;\;\frac{\sqrt{u1}}{\frac{1}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot 64.93939402268539\right)\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	9952

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

Alternative 3
Error	1.3
Cost	6692

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.5199999809265137:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot 64.93939402268539\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	6688

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

Alternative 5
Error	2.8
Cost	3808

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot 64.93939402268539\right)\right) \]

Alternative 6
Error	5.3
Cost	3624

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{elif}\;u2 \leq 0.75:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1}\\ \end{array} \]

Alternative 7
Error	5.3
Cost	3624

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\frac{u1 + u1 \cdot u1}{1 - u1 \cdot u1}}\\ \mathbf{elif}\;u2 \leq 0.75:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1}\\ \end{array} \]

Alternative 8
Error	3.9
Cost	3616

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

Alternative 9
Error	6.6
Cost	3360

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

Alternative 10
Error	11.8
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023046 
(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))))