Trowbridge-Reitz Sample, near normal, slope_x

Average Accuracy: 99.0% → 99.0%

Time: 15.7s

Precision: binary32

Cost: 10336

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1 \cdot u1}\\ \sqrt{t_0 + u1 \cdot t_0} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \end{array} \]

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 (* u1 u1)))))
   (* (sqrt (+ t_0 (* u1 t_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) {
	float t_0 = u1 / (1.0f - (u1 * u1));
	return sqrtf((t_0 + (u1 * t_0))) * 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
    real(4) :: t_0
    t_0 = u1 / (1.0e0 - (u1 * u1))
    code = sqrt((t_0 + (u1 * t_0))) * 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)
	t_0 = Float32(u1 / Float32(Float32(1.0) - Float32(u1 * u1)))
	return Float32(sqrt(Float32(t_0 + Float32(u1 * t_0))) * cos(sqrt(Float32(Float32(u2 * 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)
	t_0 = u1 / (single(1.0) - (u1 * u1));
	tmp = sqrt((t_0 + (u1 * t_0))) * cos(sqrt(((u2 * u2) * single(39.47841760436263))));
end

Derivation

1. Initial program 99.0%

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

2. Applied egg-rr 99.0%

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

   [Start] 99.0	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   flip-- [=>] 98.9	\[ \sqrt{\frac{u1}{\color{blue}{\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   associate-/r/ [=>] 98.8	\[ \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot \left(1 + u1\right)}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   +-commutative [=>] 98.8	\[ \sqrt{\frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   distribute-lft-in [=>] 99.0	\[ \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot 1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   metadata-eval [=>] 99.0	\[ \sqrt{\frac{u1}{\color{blue}{1} - u1 \cdot u1} \cdot u1 + \frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot 1} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   metadata-eval [=>] 99.0	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{\color{blue}{1} - u1 \cdot u1} \cdot 1} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

3. Applied egg-rr 99.0%

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

   [Start] 99.0	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 - u1 \cdot u1} \cdot 1} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   add-sqr-sqrt [=>] 98.8	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 - u1 \cdot u1} \cdot 1} \cdot \cos \color{blue}{\left(\sqrt{6.28318530718 \cdot u2} \cdot \sqrt{6.28318530718 \cdot u2}\right)} \]
   sqrt-unprod [=>] 99.0	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 - u1 \cdot u1} \cdot 1} \cdot \cos \color{blue}{\left(\sqrt{\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)}\right)} \]
   swap-sqr [=>] 98.9	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 - u1 \cdot u1} \cdot 1} \cdot \cos \left(\sqrt{\color{blue}{\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)}}\right) \]
   metadata-eval [=>] 99.0	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 - u1 \cdot u1} \cdot 1} \cdot \cos \left(\sqrt{\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)}\right) \]

4. Simplified 99.0%

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

   [Start] 99.0	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 - u1 \cdot u1} \cdot 1} \cdot \cos \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   *-commutative [=>] 99.0	\[ \sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot u1 + \frac{u1}{1 - u1 \cdot u1} \cdot 1} \cdot \cos \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right) \]

5. Final simplification 99.0%

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

Alternatives

Alternative 1
Accuracy	96.3%
Cost	10020

\[\begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9988020062446594:\\ \;\;\;\;t_0 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \]

Alternative 2
Accuracy	96.3%
Cost	10020

\[\begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9988020062446594:\\ \;\;\;\;t_0 \cdot \sqrt{u1 + u1 \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \]

Alternative 3
Accuracy	99.0%
Cost	7072

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1 \cdot u1}\\ \sqrt{t_0 + u1 \cdot t_0} \cdot \cos \left(u2 \cdot 6.28318530718\right) \end{array} \]

Alternative 4
Accuracy	82.9%
Cost	6820

\[\begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999200105667114:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]

Alternative 5
Accuracy	94.2%
Cost	6692

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.18000000715255737:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 6
Accuracy	99.0%
Cost	6688

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

Alternative 7
Accuracy	85.1%
Cost	3812

\[\begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.003800000064074993:\\ \;\;\;\;\sqrt{u1 + u1 \cdot u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1 \cdot \left(u1 + 1\right)}{1 - u1 \cdot u1}}\\ \end{array} \]

Alternative 8
Accuracy	88.1%
Cost	3616

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

Alternative 9
Accuracy	79.6%
Cost	3360

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

Alternative 10
Accuracy	62.9%
Cost	3232

\[\sqrt{u1} \]

Alternative 11
Accuracy	19.0%
Cost	32

\[u1 \]

Reproduce

herbie shell --seed 2023143 
(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))))