Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.5% → 99.0%
Time: 14.8s
Alternatives: 10
Speedup: 4.0×

Specification

?
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

\begin{array}{l}

\\
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
\end{array}

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 10 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 57.5% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

\begin{array}{l}

\\
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
\end{array}

Alternative 1: 99.0% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-t_0, \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot u2\right)\right)\right), {t_0}^{2}\right)\right) \end{array} \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))))
   (*
    (sqrt (- (log1p (- u1))))
    (+
     (cos (* PI (* 2.0 u2)))
     (fma (- t_0) (sin (log1p (expm1 (* PI u2)))) (pow t_0 2.0))))))
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((((float) M_PI) * u2));
	return sqrtf(-log1pf(-u1)) * (cosf((((float) M_PI) * (2.0f * u2))) + fmaf(-t_0, sinf(log1pf(expm1f((((float) M_PI) * u2)))), powf(t_0, 2.0f)));
}

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(pi) * u2))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) + fma(Float32(-t_0), sin(log1p(expm1(Float32(Float32(pi) * u2)))), (t_0 ^ Float32(2.0)))))
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sin \left(\pi \cdot u2\right)\\
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-t_0, \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot u2\right)\right)\right), {t_0}^{2}\right)\right)
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot u2\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right) \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (+
   (cos (* PI (* 2.0 u2)))
   (fma
    (- (sin (* PI u2)))
    (sin (log1p (expm1 (* PI u2))))
    (- 0.5 (* 0.5 (cos (* u2 (* PI 2.0)))))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (cosf((((float) M_PI) * (2.0f * u2))) + fmaf(-sinf((((float) M_PI) * u2)), sinf(log1pf(expm1f((((float) M_PI) * u2)))), (0.5f - (0.5f * cosf((u2 * (((float) M_PI) * 2.0f)))))));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) + fma(Float32(-sin(Float32(Float32(pi) * u2))), sin(log1p(expm1(Float32(Float32(pi) * u2)))), Float32(Float32(0.5) - Float32(Float32(0.5) * cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0)))))))))
end

\begin{array}{l}

\\
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot u2\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 99.0% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-t_0, t_0, {t_0}^{2}\right)\right) \end{array} \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))))
   (*
    (sqrt (- (log1p (- u1))))
    (+ (cos (* PI (* 2.0 u2))) (fma (- t_0) t_0 (pow t_0 2.0))))))
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((((float) M_PI) * u2));
	return sqrtf(-log1pf(-u1)) * (cosf((((float) M_PI) * (2.0f * u2))) + fmaf(-t_0, t_0, powf(t_0, 2.0f)));
}

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(pi) * u2))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) + fma(Float32(-t_0), t_0, (t_0 ^ Float32(2.0)))))
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sin \left(\pi \cdot u2\right)\\
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-t_0, t_0, {t_0}^{2}\right)\right)
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 99.0% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-t_0, t_0, 0.5 - 0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right) \end{array} \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))))
   (*
    (sqrt (- (log1p (- u1))))
    (+
     (cos (* PI (* 2.0 u2)))
     (fma (- t_0) t_0 (- 0.5 (* 0.5 (cos (* u2 (* PI 2.0))))))))))
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((((float) M_PI) * u2));
	return sqrtf(-log1pf(-u1)) * (cosf((((float) M_PI) * (2.0f * u2))) + fmaf(-t_0, t_0, (0.5f - (0.5f * cosf((u2 * (((float) M_PI) * 2.0f)))))));
}

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(pi) * u2))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) + fma(Float32(-t_0), t_0, Float32(Float32(0.5) - Float32(Float32(0.5) * cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0)))))))))
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sin \left(\pi \cdot u2\right)\\
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-t_0, t_0, 0.5 - 0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right)
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 94.7% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999991059303284:\\ \;\;\;\;t_0 \cdot \sqrt{\left(-u1\right) \cdot \left(-1 + u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* PI 2.0)))))
   (if (<= t_0 0.9999991059303284)
     (* t_0 (sqrt (* (- u1) (+ -1.0 (* u1 -0.5)))))
     (sqrt (- (log1p (- u1)))))))
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (((float) M_PI) * 2.0f)));
	float tmp;
	if (t_0 <= 0.9999991059303284f) {
		tmp = t_0 * sqrtf((-u1 * (-1.0f + (u1 * -0.5f))));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999991059303284))
		tmp = Float32(t_0 * sqrt(Float32(Float32(-u1) * Float32(Float32(-1.0) + Float32(u1 * Float32(-0.5))))));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\
\mathbf{if}\;t_0 \leq 0.9999991059303284:\\
\;\;\;\;t_0 \cdot \sqrt{\left(-u1\right) \cdot \left(-1 + u1 \cdot -0.5\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 90.5% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.0031999999191612005:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.0031999999191612005)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt u1)))))
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.0031999999191612005f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf(u1);
	}
	return tmp;
}

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0031999999191612005))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(u1));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := u2 \cdot \left(\pi \cdot 2\right)\\
\mathbf{if}\;t_0 \leq 0.0031999999191612005:\\
\;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\

\mathbf{else}:\\
\;\;\;\;\cos t_0 \cdot \sqrt{u1}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (cos (* u2 (* PI 2.0)))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((u2 * (((float) M_PI) * 2.0f)));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0)))))
end

\begin{array}{l}

\\
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 8: 80.0% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]
(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log1p (- u1)))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1));
}

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log1p(Float32(-u1))))
end

\begin{array}{l}

\\
\sqrt{-\mathsf{log1p}\left(-u1\right)}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 9: 72.8% accurate, 3.8× speedup?

\[\begin{array}{l} \\ \sqrt{\left(-u1\right) \cdot \left(-1 + u1 \cdot -0.5\right)} \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* (- u1) (+ -1.0 (* u1 -0.5)))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((-u1 * (-1.0f + (u1 * -0.5f))));
}

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 * (-0.5e0)))))
end function

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(-u1) * Float32(Float32(-1.0) + Float32(u1 * Float32(-0.5)))))
end

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((-u1 * (single(-1.0) + (u1 * single(-0.5)))));
end

\begin{array}{l}

\\
\sqrt{\left(-u1\right) \cdot \left(-1 + u1 \cdot -0.5\right)}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 10: 65.1% accurate, 4.0× speedup?

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]
(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

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)
end function

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

\begin{array}{l}

\\
\sqrt{u1}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023350 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann 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 (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))