Given's Rotation SVD example, simplified

Percentage Accurate: 75.4% → 99.8%

Time: 8.0s

Alternatives: 9

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x)))));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x)))));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 - t\_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{t\_1}^{2} - 1}{-1 - \sqrt{t\_1}}}{1.5 - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))) (t_1 (- 0.5 t_0)))
   (if (<= (hypot 1.0 x) 2.0)
     (*
      (*
       (fma
        (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
        (* x x)
        0.125)
       x)
      x)
     (/ (/ (- (pow t_1 2.0) 1.0) (- -1.0 (sqrt t_1))) (- 1.5 t_0)))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = 0.5 - t_0;
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = ((pow(t_1, 2.0) - 1.0) / (-1.0 - sqrt(t_1))) / (1.5 - t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 - t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(Float64((t_1 ^ 2.0) - 1.0) / Float64(-1.0 - sqrt(t_1))) / Float64(1.5 - t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - 1.0), $MachinePrecision] / N[(-1.0 - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 57.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} - 1\right) \cdot {\left(\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)\right)}^{-1}}{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1}} \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} - 1}{-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(t\_0 - 1\right)}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 2.0)
     (*
      (*
       (fma
        (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
        (* x x)
        0.125)
       x)
      x)
     (/ (- (- t_0 1.0)) (+ (sqrt t_0) 1.0)))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = -(t_0 - 1.0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(-Float64(t_0 - 1.0)) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[((-N[(t$95$0 - 1.0), $MachinePrecision]) / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 57.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1\right)}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (*
     (fma
      (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
      (* x x)
      0.125)
     x)
    x)
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 57.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\sqrt{\frac{1}{2}} + 1}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\sqrt{\frac{1}{2}} + 1}} \]

      4. lower-sqrt.f64 99.0

         \[\leadsto \frac{0.5}{\color{blue}{\sqrt{0.5}} + 1} \]

   7. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{0.5} + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(0.0673828125 * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 57.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2}} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      9. sub-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{69}{1024} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{11}{128}\right)\right)}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} + \color{blue}{\frac{-11}{128}}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{69}{1024}, {x}^{2}, \frac{-11}{128}\right)}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, \color{blue}{x \cdot x}, \frac{-11}{128}\right), {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, \color{blue}{x \cdot x}, \frac{-11}{128}\right), {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, x \cdot x, \frac{-11}{128}\right), \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      15. lower-*.f64 99.8

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\sqrt{\frac{1}{2}} + 1}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\sqrt{\frac{1}{2}} + 1}} \]

      4. lower-sqrt.f64 99.0

         \[\leadsto \frac{0.5}{\color{blue}{\sqrt{0.5}} + 1} \]

   7. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{0.5} + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 57.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 99.7

         \[\leadsto \left(\mathsf{fma}\left(-0.0859375, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\sqrt{\frac{1}{2}} + 1}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\sqrt{\frac{1}{2}} + 1}} \]

      4. lower-sqrt.f64 99.0

         \[\leadsto \frac{0.5}{\color{blue}{\sqrt{0.5}} + 1} \]

   7. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{0.5} + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)
   (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 57.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 99.7

         \[\leadsto \left(\mathsf{fma}\left(-0.0859375, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(0.125 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0) (* (* 0.125 x) x) (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (0.125 * x) * x;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = (0.125 * x) * x;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = (0.125 * x) * x
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(0.125 * x) * x);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (0.125 * x) * x;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 57.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2}} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      9. sub-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{69}{1024} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{11}{128}\right)\right)}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} + \color{blue}{\frac{-11}{128}}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{69}{1024}, {x}^{2}, \frac{-11}{128}\right)}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, \color{blue}{x \cdot x}, \frac{-11}{128}\right), {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, \color{blue}{x \cdot x}, \frac{-11}{128}\right), {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, x \cdot x, \frac{-11}{128}\right), \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      15. lower-*.f64 99.8

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 99.5%

       \[\leadsto \left(0.125 \cdot x\right) \cdot x \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \left(0.125 \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 0.125 x) x))

C

double code(double x) {
	return (0.125 * x) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.125d0 * x) * x
end function

Java

public static double code(double x) {
	return (0.125 * x) * x;
}

Python

def code(x):
	return (0.125 * x) * x

Julia

function code(x)
	return Float64(Float64(0.125 * x) * x)
end

MATLAB

function tmp = code(x)
	tmp = (0.125 * x) * x;
end

Wolfram

code[x_] := N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Add Preprocessing

4. Applied rewrites 77.5%

   \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2}} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

   9. sub-neg N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{69}{1024} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{11}{128}\right)\right)}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

   10. metadata-eval N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} + \color{blue}{\frac{-11}{128}}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{69}{1024}, {x}^{2}, \frac{-11}{128}\right)}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

   12. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, \color{blue}{x \cdot x}, \frac{-11}{128}\right), {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

   13. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, \color{blue}{x \cdot x}, \frac{-11}{128}\right), {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

   14. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{69}{1024}, x \cdot x, \frac{-11}{128}\right), \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

   15. lower-*.f64 55.0

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

7. Applied rewrites 55.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 55.3%

    \[\leadsto \left(0.125 \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 9: 52.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.125 (* x x)))

C

double code(double x) {
	return 0.125 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.125d0 * (x * x)
end function

Java

public static double code(double x) {
	return 0.125 * (x * x);
}

Python

def code(x):
	return 0.125 * (x * x)

Julia

function code(x)
	return Float64(0.125 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 0.125 * (x * x);
end

Wolfram

code[x_] := N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Add Preprocessing

4. Applied rewrites 77.5%

   \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{8} \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. lower-*.f64 55.3

      \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

7. Applied rewrites 55.3%

   \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024299 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))