Given's Rotation SVD example, simplified

Percentage Accurate: 75.3% → 99.9%

Time: 4.8s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

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: 75.3% accurate, 1.0× speedup

Math

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

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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := \frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}}\\ \mathbf{if}\;\left|x\right| \leq 0.00054:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, t\_0, 0.125\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - 0.5}{-1 - \sqrt{t\_1 - -0.5}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x)))
        (t_1 (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0)))))
   (if (<= (fabs x) 0.00054)
     (* (fma -0.0859375 t_0 0.125) t_0)
     (/ (- t_1 0.5) (- -1.0 (sqrt (- t_1 -0.5)))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = 0.5 / sqrt(fma(fabs(x), fabs(x), 1.0));
	double tmp;
	if (fabs(x) <= 0.00054) {
		tmp = fma(-0.0859375, t_0, 0.125) * t_0;
	} else {
		tmp = (t_1 - 0.5) / (-1.0 - sqrt((t_1 - -0.5)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0)))
	tmp = 0.0
	if (abs(x) <= 0.00054)
		tmp = Float64(fma(-0.0859375, t_0, 0.125) * t_0);
	else
		tmp = Float64(Float64(t_1 - 0.5) / Float64(-1.0 - sqrt(Float64(t_1 - -0.5))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.00054], N[(N[(-0.0859375 * t$95$0 + 0.125), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$1 - 0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(t$95$1 - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.40000000000000007e-4

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.1%

         \[\leadsto {x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.1%

         \[\leadsto \left(0.125 + -0.0859375 \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 50.1%

         \[\leadsto \mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot {\color{blue}{x}}^{2} \]

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 50.1%

          \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot {x}^{2} \]

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 50.1%

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

   6. Applied rewrites 50.1%

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

   if 5.40000000000000007e-4 < x

   1. Initial program 75.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. associate--l- N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. sub-flip-reverse N/A

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

   5. Applied rewrites 76.1%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := \frac{0.5}{\left|x\right|}\\ \mathbf{if}\;\left|x\right| \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, t\_0, 0.125\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - 0.5}{-\left(\sqrt{t\_1 - -0.5} - -1\right)}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))) (t_1 (/ 0.5 (fabs x))))
   (if (<= (fabs x) 1.1)
     (* (fma -0.0859375 t_0 0.125) t_0)
     (/ (- t_1 0.5) (- (- (sqrt (- t_1 -0.5)) -1.0))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = 0.5 / fabs(x);
	double tmp;
	if (fabs(x) <= 1.1) {
		tmp = fma(-0.0859375, t_0, 0.125) * t_0;
	} else {
		tmp = (t_1 - 0.5) / -(sqrt((t_1 - -0.5)) - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(0.5 / abs(x))
	tmp = 0.0
	if (abs(x) <= 1.1)
		tmp = Float64(fma(-0.0859375, t_0, 0.125) * t_0);
	else
		tmp = Float64(Float64(t_1 - 0.5) / Float64(-Float64(sqrt(Float64(t_1 - -0.5)) - -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.1], N[(N[(-0.0859375 * t$95$0 + 0.125), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$1 - 0.5), $MachinePrecision] / (-N[(N[Sqrt[N[(t$95$1 - -0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.1%

         \[\leadsto {x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.1%

         \[\leadsto \left(0.125 + -0.0859375 \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 50.1%

         \[\leadsto \mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot {\color{blue}{x}}^{2} \]

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 50.1%

          \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot {x}^{2} \]

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 50.1%

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

   6. Applied rewrites 50.1%

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

   if 1.1000000000000001 < x

   1. Initial program 75.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 76.1%

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

   4. Taylor expanded in x around inf

      \[\leadsto \frac{1 \cdot 1 - \left(\color{blue}{\frac{\frac{1}{2}}{x}} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} - -0.5}} \]
   5. Step-by-step derivation

      1. lower-/.f64 50.4%

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

   6. Applied rewrites 50.4%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1 \cdot 1 - \left(\frac{0.5}{x} - -0.5\right)}{1 + \sqrt{\color{blue}{\frac{\frac{1}{2}}{x}} - -0.5}} \]
   8. Step-by-step derivation

      1. lower-/.f64 49.7%

         \[\leadsto \frac{1 \cdot 1 - \left(\frac{0.5}{x} - -0.5\right)}{1 + \sqrt{\frac{0.5}{\color{blue}{x}} - -0.5}} \]

   9. Applied rewrites 49.7%

      \[\leadsto \frac{1 \cdot 1 - \left(\frac{0.5}{x} - -0.5\right)}{1 + \sqrt{\color{blue}{\frac{0.5}{x}} - -0.5}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{\frac{1}{2}}{x} - \frac{-1}{2}\right)}{1 + \sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}}} \]

       2. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 \cdot 1 - \left(\frac{\frac{1}{2}}{x} - \frac{-1}{2}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + \sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right)}} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 \cdot 1 - \left(\frac{\frac{1}{2}}{x} - \frac{-1}{2}\right)\right)\right)}{\mathsf{neg}\left(\left(1 + \sqrt{\frac{\frac{1}{2}}{x} - \frac{-1}{2}}\right)\right)}} \]

   11. Applied rewrites 49.7%

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

Alternative 3: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 0.00065:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, t\_0, 0.125\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, 1\right)}} - -0.5}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (if (<= (fabs x) 0.00065)
     (* (fma -0.0859375 t_0 0.125) t_0)
     (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma (fabs x) (fabs x) 1.0))) -0.5))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 0.00065) {
		tmp = fma(-0.0859375, t_0, 0.125) * t_0;
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(fabs(x), fabs(x), 1.0))) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 0.00065)
		tmp = Float64(fma(-0.0859375, t_0, 0.125) * t_0);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(abs(x), abs(x), 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.00065], N[(N[(-0.0859375 * t$95$0 + 0.125), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.4999999999999997e-4

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.1%

         \[\leadsto {x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.1%

         \[\leadsto \left(0.125 + -0.0859375 \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 50.1%

         \[\leadsto \mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot {\color{blue}{x}}^{2} \]

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 50.1%

          \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot {x}^{2} \]

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 50.1%

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

   6. Applied rewrites 50.1%

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

   if 6.4999999999999997e-4 < x

   1. Initial program 75.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

          \[\leadsto 1 - \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\mathsf{hypot}\left(1, x\right)} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto 1 - \sqrt{\color{blue}{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      12. lift-hypot.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{\sqrt{\color{blue}{1} + x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{\sqrt{\color{blue}{x \cdot x + 1}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      17. metadata-eval 75.3%

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

   3. Applied rewrites 75.3%

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

Alternative 4: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \leq 0.8:\\ \;\;\;\;0.2928932188134525\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x))))) 0.8)
   0.2928932188134525
   (* (fma -0.0859375 (* x x) 0.125) (* x x))))

C

double code(double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))) <= 0.8) {
		tmp = 0.2928932188134525;
	} else {
		tmp = fma(-0.0859375, (x * x), 0.125) * (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))) <= 0.8)
		tmp = 0.2928932188134525;
	else
		tmp = Float64(fma(-0.0859375, Float64(x * x), 0.125) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], 0.2928932188134525, N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   1. Initial program 75.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 76.1%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 51.0%

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

   6. Applied rewrites 51.0%

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

   7. Evaluated real constant 51.0%

      \[\leadsto 0.2928932188134525 \]

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.1%

         \[\leadsto {x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.1%

         \[\leadsto \left(0.125 + -0.0859375 \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 50.1%

         \[\leadsto \mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot {\color{blue}{x}}^{2} \]

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 50.1%

          \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot {x}^{2} \]

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 50.1%

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

   6. Applied rewrites 50.1%

      \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \leq 0.8:\\ \;\;\;\;0.2928932188134525\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x))))) 0.8)
   0.2928932188134525
   (* 0.125 (* x x))))

C

double code(double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))) <= 0.8) {
		tmp = 0.2928932188134525;
	} else {
		tmp = 0.125 * (x * x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x))))) <= 0.8) {
		tmp = 0.2928932188134525;
	} else {
		tmp = 0.125 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x))))) <= 0.8:
		tmp = 0.2928932188134525
	else:
		tmp = 0.125 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))) <= 0.8)
		tmp = 0.2928932188134525;
	else
		tmp = Float64(0.125 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))) <= 0.8)
		tmp = 0.2928932188134525;
	else
		tmp = 0.125 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], 0.2928932188134525, N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   1. Initial program 75.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 76.1%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 51.0%

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

   6. Applied rewrites 51.0%

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

   7. Evaluated real constant 51.0%

      \[\leadsto 0.2928932188134525 \]

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 50.1%

         \[\leadsto {x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.1%

         \[\leadsto \left(0.125 + -0.0859375 \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 50.1%

         \[\leadsto \mathsf{fma}\left(-0.0859375, {x}^{2}, 0.125\right) \cdot {\color{blue}{x}}^{2} \]

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 50.1%

          \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot {x}^{2} \]

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 50.1%

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

   6. Applied rewrites 50.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 51.6%

      \[\leadsto 0.125 \cdot \left(\color{blue}{x} \cdot x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.12 \cdot 10^{-79}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;0.2928932188134525\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1.12e-79) (- 1.0 1.0) 0.2928932188134525))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.12e-79) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 1.12d-79) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 0.2928932188134525d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1.12e-79) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 0.2928932188134525;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1.12e-79:
		tmp = 1.0 - 1.0
	else:
		tmp = 0.2928932188134525
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.12e-79)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = 0.2928932188134525;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1.12e-79)
		tmp = 1.0 - 1.0;
	else
		tmp = 0.2928932188134525;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1.12e-79], N[(1.0 - 1.0), $MachinePrecision], 0.2928932188134525]

Derivation

1. Split input into 2 regimes

2. if x < 1.11999999999999996e-79

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 26.8%

      \[\leadsto 1 - \color{blue}{1} \]

   if 1.11999999999999996e-79 < x

   1. Initial program 75.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 76.1%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 51.0%

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

   6. Applied rewrites 51.0%

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

   7. Evaluated real constant 51.0%

      \[\leadsto 0.2928932188134525 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.0% accurate, 27.4× speedup

Math

\[0.2928932188134525 \]

FPCore

(FPCore (x) :precision binary64 0.2928932188134525)

C

double code(double x) {
	return 0.2928932188134525;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.2928932188134525d0
end function

Java

public static double code(double x) {
	return 0.2928932188134525;
}

Python

def code(x):
	return 0.2928932188134525

Julia

function code(x)
	return 0.2928932188134525
end

MATLAB

function tmp = code(x)
	tmp = 0.2928932188134525;
end

Wolfram

code[x_] := 0.2928932188134525

Derivation

1. Initial program 75.3%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-unsound-/.f64 N/A

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

3. Applied rewrites 76.1%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 51.0%

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

6. Applied rewrites 51.0%

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

7. Evaluated real constant 51.0%

   \[\leadsto 0.2928932188134525 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))