Trowbridge-Reitz Sample, near normal, slope_y

Average Error: 0.5 → 0.5

Time: 1.5min

Precision: binary64

Cost: 20612

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{u1}{u1 - 1}\\ t_1 := \frac{u1}{1 - u1}\\ \sqrt{\begin{array}{l} \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;\frac{{t_0}^{2}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary64
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary64
 (let* ((t_0 (/ u1 (- u1 1.0))) (t_1 (/ u1 (- 1.0 u1))))
   (*
    (sqrt (if (!= t_0 0.0) (/ (pow t_0 2.0) t_1) t_1))
    (sin (* 6.28318530718 u2)))))

C

double code(double cosTheta_i, double u1, double u2) {
	return sqrt((u1 / (1.0 - u1))) * sin((6.28318530718 * u2));
}

↓

double code(double cosTheta_i, double u1, double u2) {
	double t_0 = u1 / (u1 - 1.0);
	double t_1 = u1 / (1.0 - u1);
	double tmp;
	if (t_0 != 0.0) {
		tmp = pow(t_0, 2.0) / t_1;
	} else {
		tmp = t_1;
	}
	return sqrt(tmp) * sin((6.28318530718 * u2));
}

Fortran

real(8) function code(costheta_i, u1, u2)
    real(8), intent (in) :: costheta_i
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt((u1 / (1.0d0 - u1))) * sin((6.28318530718d0 * u2))
end function

↓

real(8) function code(costheta_i, u1, u2)
    real(8), intent (in) :: costheta_i
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u1 / (u1 - 1.0d0)
    t_1 = u1 / (1.0d0 - u1)
    if (t_0 /= 0.0d0) then
        tmp = (t_0 ** 2.0d0) / t_1
    else
        tmp = t_1
    end if
    code = sqrt(tmp) * sin((6.28318530718d0 * u2))
end function

Java

public static double code(double cosTheta_i, double u1, double u2) {
	return Math.sqrt((u1 / (1.0 - u1))) * Math.sin((6.28318530718 * u2));
}

↓

public static double code(double cosTheta_i, double u1, double u2) {
	double t_0 = u1 / (u1 - 1.0);
	double t_1 = u1 / (1.0 - u1);
	double tmp;
	if (t_0 != 0.0) {
		tmp = Math.pow(t_0, 2.0) / t_1;
	} else {
		tmp = t_1;
	}
	return Math.sqrt(tmp) * Math.sin((6.28318530718 * u2));
}

Python

def code(cosTheta_i, u1, u2):
	return math.sqrt((u1 / (1.0 - u1))) * math.sin((6.28318530718 * u2))

↓

def code(cosTheta_i, u1, u2):
	t_0 = u1 / (u1 - 1.0)
	t_1 = u1 / (1.0 - u1)
	tmp = 0
	if t_0 != 0.0:
		tmp = math.pow(t_0, 2.0) / t_1
	else:
		tmp = t_1
	return math.sqrt(tmp) * math.sin((6.28318530718 * u2))

Julia

function code(cosTheta_i, u1, u2)
	return Float64(sqrt(Float64(u1 / Float64(1.0 - u1))) * sin(Float64(6.28318530718 * u2)))
end

↓

function code(cosTheta_i, u1, u2)
	t_0 = Float64(u1 / Float64(u1 - 1.0))
	t_1 = Float64(u1 / Float64(1.0 - u1))
	tmp = 0.0
	if (t_0 != 0.0)
		tmp = Float64((t_0 ^ 2.0) / t_1);
	else
		tmp = t_1;
	end
	return Float64(sqrt(tmp) * sin(Float64(6.28318530718 * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (1.0 - u1))) * sin((6.28318530718 * u2));
end

↓

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u1 / (u1 - 1.0);
	t_1 = u1 / (1.0 - u1);
	tmp = 0.0;
	if (t_0 ~= 0.0)
		tmp = (t_0 ^ 2.0) / t_1;
	else
		tmp = t_1;
	end
	tmp_2 = sqrt(tmp) * sin((6.28318530718 * u2));
end

Wolfram

code[cosTheta$95$i_, u1_, u2_] := N[(N[Sqrt[N[(u1 / N[(1.0 - u1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(6.28318530718 * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[cosTheta$95$i_, u1_, u2_] := Block[{t$95$0 = N[(u1 / N[(u1 - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(u1 / N[(1.0 - u1), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[If[Unequal[t$95$0, 0.0], N[(N[Power[t$95$0, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision], t$95$1]], $MachinePrecision] * N[Sin[N[(6.28318530718 * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

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

   \[\leadsto \sqrt{\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{0}{u1 + -1} + \frac{u1}{u1 + -1} \ne 0:\\ \;\;\;\;\frac{\frac{0}{u1 + -1} \cdot \frac{0}{u1 + -1} - {\left(\frac{u1}{1 - u1}\right)}^{2}}{\frac{0}{u1 + -1} + \frac{u1}{u1 + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u1}{1 - u1}\\ } \end{array}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Simplified 0.5

   \[\leadsto \sqrt{\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{u1}{u1 - 1} \ne 0:\\ \;\;\;\;\frac{{\left(\frac{u1}{u1 - 1}\right)}^{2}}{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u1}{1 - u1}\\ } \end{array}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   Proof

Alternatives

Alternative 1
Error	0.5
Cost	14532

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \sqrt{\begin{array}{l} \mathbf{if}\;\frac{u1}{u1 - 1} \ne 0:\\ \;\;\;\;\frac{u1 \cdot \frac{t_0}{1 - u1}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

Alternative 2
Error	14.6
Cost	13636

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \begin{array}{l} \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;\frac{1}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0}\\ \end{array}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

Alternative 3
Error	21.0
Cost	13380

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.01055:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \begin{array}{l} \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;\frac{1}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0}\\ \end{array}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	13376

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

Alternative 5
Error	24.4
Cost	7556

\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ 6.28318530718 \cdot \left(u2 \cdot \begin{array}{l} \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;\frac{1}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0}\\ \end{array}\right) \end{array} \]

Alternative 6
Error	24.4
Cost	7104

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

Alternative 7
Error	24.4
Cost	6976

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

Alternative 8
Error	24.4
Cost	6976

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

Alternative 9
Error	39.4
Cost	6720

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

Alternative 10
Error	39.4
Cost	6720

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

Alternative 11
Error	39.4
Cost	6720

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

Reproduce

herbie shell --seed 2023033 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary64
  :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))))