bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.6%

Time: 22.9s

Alternatives: 11

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 7.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot 0.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := {t_1}^{0.16666666666666666}\\ t_3 := -t_2\\ t_4 := t_2 \cdot {\left(\sqrt[3]{t_1}\right)}^{2}\\ \mathsf{fma}\left(t_0, t_0, t_4 \cdot t_3\right) + \mathsf{fma}\left(t_3, t_4, t_2 \cdot t_4\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (* PI 0.5)))
        (t_1 (asin (- 1.0 x)))
        (t_2 (pow t_1 0.16666666666666666))
        (t_3 (- t_2))
        (t_4 (* t_2 (pow (cbrt t_1) 2.0))))
   (+ (fma t_0 t_0 (* t_4 t_3)) (fma t_3 t_4 (* t_2 t_4)))))

C

double code(double x) {
	double t_0 = sqrt((((double) M_PI) * 0.5));
	double t_1 = asin((1.0 - x));
	double t_2 = pow(t_1, 0.16666666666666666);
	double t_3 = -t_2;
	double t_4 = t_2 * pow(cbrt(t_1), 2.0);
	return fma(t_0, t_0, (t_4 * t_3)) + fma(t_3, t_4, (t_2 * t_4));
}

Julia

function code(x)
	t_0 = sqrt(Float64(pi * 0.5))
	t_1 = asin(Float64(1.0 - x))
	t_2 = t_1 ^ 0.16666666666666666
	t_3 = Float64(-t_2)
	t_4 = Float64(t_2 * (cbrt(t_1) ^ 2.0))
	return Float64(fma(t_0, t_0, Float64(t_4 * t_3)) + fma(t_3, t_4, Float64(t_2 * t_4)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 0.16666666666666666], $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(t$95$2 * N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * t$95$0 + N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * t$95$4 + N[(t$95$2 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sin^{-1} \left(1 - x\right)}\\ \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-t_0, t_0, {t_0}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (asin (- 1.0 x)))))
   (+ (acos (- 1.0 x)) (fma (- t_0) t_0 (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = sqrt(asin((1.0 - x)));
	return acos((1.0 - x)) + fma(-t_0, t_0, pow(t_0, 2.0));
}

Julia

function code(x)
	t_0 = sqrt(asin(Float64(1.0 - x)))
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_0), t_0, (t_0 ^ 2.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$0) * t$95$0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 3: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \mathsf{fma}\left(-t_1, t_1, t_0\right) + \left(\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+ (fma (- t_1) t_1 t_0) (+ (+ 1.0 (acos (- 1.0 x))) -1.0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return fma(-t_1, t_1, t_0) + ((1.0 + acos((1.0 - x))) + -1.0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(fma(Float64(-t_1), t_1, t_0) + Float64(Float64(1.0 + acos(Float64(1.0 - x))) + -1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision] + N[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 4: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \left(1 + \left(\cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-t_1, t_1, t_0\right)\right)\right) + -1 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+ (+ 1.0 (+ (acos (- 1.0 x)) (fma (- t_1) t_1 t_0))) -1.0)))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return (1.0 + (acos((1.0 - x)) + fma(-t_1, t_1, t_0))) + -1.0;
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(Float64(1.0 + Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_1), t_1, t_0))) + -1.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(1.0 + N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 5: 10.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-t_1, t_1, t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+ (acos (- 1.0 x)) (fma (- t_1) t_1 t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return acos((1.0 - x)) + fma(-t_1, t_1, t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(acos(Float64(1.0 - x)) + fma(Float64(-t_1), t_1, t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 6: 10.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ \pi \cdot 0.5 - t_0 \cdot {t_0}^{2} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x))))) (- (* PI 0.5) (* t_0 (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	return (((double) M_PI) * 0.5) - (t_0 * pow(t_0, 2.0));
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.asin((1.0 - x)));
	return (Math.PI * 0.5) - (t_0 * Math.pow(t_0, 2.0));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	return Float64(Float64(pi * 0.5) - Float64(t_0 * (t_0 ^ 2.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(Pi * 0.5), $MachinePrecision] - N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 7: 9.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \sqrt{{\left(t_0 + -1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{t_0}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17)
     (+ 1.0 (sqrt (pow (+ t_0 -1.0) 2.0)))
     (cbrt (pow t_0 3.0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + sqrt(pow((t_0 + -1.0), 2.0));
	} else {
		tmp = cbrt(pow(t_0, 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + Math.sqrt(Math.pow((t_0 + -1.0), 2.0));
	} else {
		tmp = Math.cbrt(Math.pow(t_0, 3.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(1.0 + sqrt((Float64(t_0 + -1.0) ^ 2.0)));
	else
		tmp = cbrt((t_0 ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(1.0 + N[Sqrt[N[Power[N[(t$95$0 + -1.0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 8: 9.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\cos^{-1} \left(1 - x\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (+ (* PI 0.5) (asin (- 1.0 x)))
   (cbrt (pow (acos (- 1.0 x)) 3.0))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = cbrt(pow(acos((1.0 - x)), 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = Math.cbrt(Math.pow(Math.acos((1.0 - x)), 3.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = cbrt((acos(Float64(1.0 - x)) ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 9: 9.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\cos^{-1} \left(1 - x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (+ (* PI 0.5) (asin (- 1.0 x)))
   (/ 1.0 (/ 1.0 (acos (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = 1.0 / (1.0 / acos((1.0 - x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = 1.0 / (1.0 / Math.acos((1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	else:
		tmp = 1.0 / (1.0 / math.acos((1.0 - x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = Float64(1.0 / Float64(1.0 / acos(Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = (pi * 0.5) + asin((1.0 - x));
	else
		tmp = 1.0 / (1.0 / acos((1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 7.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\cos^{-1} \left(1 - x\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.0 (acos (- 1.0 x)))))

C

double code(double x) {
	return 1.0 / (1.0 / acos((1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 / acos((1.0d0 - x)))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 / Math.acos((1.0 - x)));
}

Python

def code(x):
	return 1.0 / (1.0 / math.acos((1.0 - x)))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 / acos(Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 / acos((1.0 - x)));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 / N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 7.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Derivation

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1. Add Preprocessing

Developer target: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (asin (sqrt (/ x 2.0)))))

C

double code(double x) {
	return 2.0 * asin(sqrt((x / 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * asin(sqrt((x / 2.0d0)))
end function

Java

public static double code(double x) {
	return 2.0 * Math.asin(Math.sqrt((x / 2.0)));
}

Python

def code(x):
	return 2.0 * math.asin(math.sqrt((x / 2.0)))

Julia

function code(x)
	return Float64(2.0 * asin(sqrt(Float64(x / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * asin(sqrt((x / 2.0)));
end

Wolfram

code[x_] := N[(2.0 * N[ArcSin[N[Sqrt[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023350 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))