Trowbridge-Reitz Sample, near normal, slope_y

Average Accuracy: 98.4% → 98.4%

Time: 15.0s

Precision: binary32

Cost: 9952

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

Math

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

↓

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

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/ (sin (sqrt (* (* u2 u2) 39.47841760436263))) (sqrt (- -1.0 (/ -1.0 u1)))))

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

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 = sin(sqrt(((u2 * u2) * 39.47841760436263e0))) / sqrt(((-1.0e0) - ((-1.0e0) / u1)))
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(sin(sqrt(Float32(Float32(u2 * u2) * Float32(39.47841760436263)))) / sqrt(Float32(Float32(-1.0) - Float32(Float32(-1.0) / u1))))
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 = sin(sqrt(((u2 * u2) * single(39.47841760436263)))) / sqrt((single(-1.0) - (single(-1.0) / u1)));
end

Derivation

1. Initial program 98.4%

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

2. Applied egg-rr 98.5%

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

   [Start] 98.4	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   add-sqr-sqrt [=>] 97.7	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{6.28318530718 \cdot u2} \cdot \sqrt{6.28318530718 \cdot u2}\right)} \]
   sqrt-unprod [=>] 98.4	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)}\right)} \]
   swap-sqr [=>] 98.2	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\color{blue}{\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)}}\right) \]
   metadata-eval [=>] 98.5	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)}\right) \]

3. Applied egg-rr 98.4%

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

   [Start] 98.5	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   clear-num [=>] 98.4	\[ \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   sqrt-div [=>] 98.3	\[ \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   metadata-eval [=>] 98.3	\[ \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   associate-*l/ [=>] 98.4	\[ \color{blue}{\frac{1 \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]
   *-un-lft-identity [<=] 98.4	\[ \frac{\color{blue}{\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]
   sqrt-prod [=>] 97.9	\[ \frac{\sin \color{blue}{\left(\sqrt{39.47841760436263} \cdot \sqrt{u2 \cdot u2}\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]
   *-commutative [=>] 97.9	\[ \frac{\sin \color{blue}{\left(\sqrt{u2 \cdot u2} \cdot \sqrt{39.47841760436263}\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]
   sqrt-prod [=>] 97.5	\[ \frac{\sin \left(\color{blue}{\left(\sqrt{u2} \cdot \sqrt{u2}\right)} \cdot \sqrt{39.47841760436263}\right)}{\sqrt{\frac{1 - u1}{u1}}} \]
   add-sqr-sqrt [<=] 97.9	\[ \frac{\sin \left(\color{blue}{u2} \cdot \sqrt{39.47841760436263}\right)}{\sqrt{\frac{1 - u1}{u1}}} \]
   metadata-eval [=>] 98.4	\[ \frac{\sin \left(u2 \cdot \color{blue}{6.28318530718}\right)}{\sqrt{\frac{1 - u1}{u1}}} \]

4. Simplified 98.3%

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

   [Start] 98.4	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1 - u1}{u1}}} \]
   div-sub [=>] 98.3	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]
   *-inverses [=>] 98.3	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1} - \color{blue}{1}}} \]
   sub-neg [=>] 98.3	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]
   metadata-eval [=>] 98.3	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

5. Applied egg-rr 98.4%

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

   [Start] 98.3	\[ \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}} \]
   add-sqr-sqrt [=>] 97.6	\[ \frac{\sin \color{blue}{\left(\sqrt{u2 \cdot 6.28318530718} \cdot \sqrt{u2 \cdot 6.28318530718}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
   sqrt-unprod [=>] 98.3	\[ \frac{\sin \color{blue}{\left(\sqrt{\left(u2 \cdot 6.28318530718\right) \cdot \left(u2 \cdot 6.28318530718\right)}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
   swap-sqr [=>] 98.1	\[ \frac{\sin \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)}}\right)}{\sqrt{\frac{1}{u1} + -1}} \]
   metadata-eval [=>] 98.4	\[ \frac{\sin \left(\sqrt{\left(u2 \cdot u2\right) \cdot \color{blue}{39.47841760436263}}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

6. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	98.5%
Cost	9952

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

Alternative 2
Accuracy	94.3%
Cost	6692

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.19499999284744263:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3
Accuracy	98.4%
Cost	6688

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

Alternative 4
Accuracy	98.3%
Cost	6688

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

Alternative 5
Accuracy	98.4%
Cost	6688

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

Alternative 6
Accuracy	89.4%
Cost	3680

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \]

Alternative 7
Accuracy	82.2%
Cost	3552

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

Alternative 8
Accuracy	82.2%
Cost	3552

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

Alternative 9
Accuracy	82.2%
Cost	3552

\[\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u2 \cdot u1}}} \]

Alternative 10
Accuracy	81.8%
Cost	3488

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

Alternative 11
Accuracy	81.8%
Cost	3488

\[6.28318530718 \cdot \frac{u2}{\sqrt{-1 - \frac{-1}{u1}}} \]

Alternative 12
Accuracy	81.8%
Cost	3488

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

Alternative 13
Accuracy	82.1%
Cost	3488

\[u2 \cdot \sqrt{\frac{39.47841760436263}{-1 - \frac{-1}{u1}}} \]

Alternative 14
Accuracy	4.6%
Cost	3360

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

Alternative 15
Accuracy	64.9%
Cost	3360

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

Alternative 16
Accuracy	64.9%
Cost	3360

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

Reproduce

herbie shell --seed 2023135 
(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))))