Given's Rotation SVD example, simplified

Percentage Accurate: 75.3% → 99.9%

Time: 5.0s

Alternatives: 12

Speedup: 2.6×

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: 75.3% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\\ t_1 := t\_0 \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.0285:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_1}^{1.5}}{1 + \mathsf{fma}\left(t\_0, 0.5, 1 \cdot \sqrt{t\_1}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0)) (t_1 (* t_0 0.5)))
   (if (<= x_m 0.0285)
     (*
      (fma
       (fma
        (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
        (* x_m x_m)
        -0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (/ (- 1.0 (pow t_1 1.5)) (+ 1.0 (fma t_0 0.5 (* 1.0 (sqrt t_1))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0;
	double t_1 = t_0 * 0.5;
	double tmp;
	if (x_m <= 0.0285) {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - pow(t_1, 1.5)) / (1.0 + fma(t_0, 0.5, (1.0 * sqrt(t_1))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0)
	t_1 = Float64(t_0 * 0.5)
	tmp = 0.0
	if (x_m <= 0.0285)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - (t_1 ^ 1.5)) / Float64(1.0 + fma(t_0, 0.5, Float64(1.0 * sqrt(t_1)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0285], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * 0.5 + N[(1.0 * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.028500000000000001

   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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

   6. Applied rewrites 51.8%

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

   if 0.028500000000000001 < x

   1. Initial program 75.3%

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

   3. Applied rewrites 76.0%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5)))
   (if (<= x_m 0.023)
     (*
      (fma
       (fma
        (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
        (* x_m x_m)
        -0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5;
	double tmp;
	if (x_m <= 0.023) {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)
	tmp = 0.0
	if (x_m <= 0.023)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.023], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.023

   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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

   6. Applied rewrites 51.8%

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

   if 0.023 < 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. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-special-/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 76.0%

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

Alternative 3: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0305:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0305)
   (*
    (fma
     (fma
      (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
      (* x_m x_m)
      -0.0859375)
     (* x_m x_m)
     0.125)
    (* x_m x_m))
   (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma x_m x_m 1.0))) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0305) {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0305)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0305], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.030499999999999999

   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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

   6. Applied rewrites 51.8%

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

   if 0.030499999999999999 < 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0115:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0115)
   (*
    (fma (fma 0.0673828125 (* x_m x_m) -0.0859375) (* x_m x_m) 0.125)
    (* x_m x_m))
   (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma x_m x_m 1.0))) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0115) {
		tmp = fma(fma(0.0673828125, (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0115)
		tmp = Float64(fma(fma(0.0673828125, Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0115], N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0115

   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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 53.1%

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

   if 0.0115 < 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0115:\\ \;\;\;\;\left(\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0673828125 \cdot x\_m, x\_m, -0.0859375\right), 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0115)
   (*
    (* (fma (* x_m x_m) (fma (* 0.0673828125 x_m) x_m -0.0859375) 0.125) x_m)
    x_m)
   (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma x_m x_m 1.0))) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0115) {
		tmp = (fma((x_m * x_m), fma((0.0673828125 * x_m), x_m, -0.0859375), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0115)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), fma(Float64(0.0673828125 * x_m), x_m, -0.0859375), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0115], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0673828125 * x$95$m), $MachinePrecision] * x$95$m + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0115

   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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 53.1%

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

   if 0.0115 < 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

Alternative 6: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00255:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00255)
   (* (fma -0.0859375 (* x_m x_m) 0.125) (* x_m x_m))
   (- 1.0 (sqrt (- (/ 0.5 (sqrt (fma x_m x_m 1.0))) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00255) {
		tmp = fma(-0.0859375, (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00255)
		tmp = Float64(fma(-0.0859375, Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00255], N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0025500000000000002

   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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.7%

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

   if 0.0025500000000000002 < 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

Alternative 7: 98.6% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (fma -0.0859375 (* x_m x_m) 0.125) (* x_m x_m))
   (- 1.0 (sqrt (- 0.5 (/ -0.5 x_m))))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]], $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. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.7%

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

   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

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

      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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{-1}{2} \cdot \frac{\color{blue}{1}}{x}} \]

      3. mult-flip N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 48.4

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

   8. Applied rewrites 48.4%

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

Alternative 8: 98.6% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (* (fma (* -0.0859375 x_m) x_m 0.125) x_m) x_m)
   (- 1.0 (sqrt (- 0.5 (/ -0.5 x_m))))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(-0.0859375 * x$95$m), $MachinePrecision] * x$95$m + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]], $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. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 53.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 51.7%

      \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 51.7

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

       6. lift-*.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. associate-*r* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 51.7

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

   11. Applied rewrites 51.7%

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

   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

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

      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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{-1}{2} \cdot \frac{\color{blue}{1}}{x}} \]

      3. mult-flip N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 48.4

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

   8. Applied rewrites 48.4%

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

Alternative 9: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.22:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{-0.5}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.22) (* 0.125 (* x_m x_m)) (- 1.0 (sqrt (- 0.5 (/ -0.5 x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.22) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt((0.5 - (-0.5 / x_m)));
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.22d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 1.0d0 - sqrt((0.5d0 - ((-0.5d0) / x_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.22) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - Math.sqrt((0.5 - (-0.5 / x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.22:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = 1.0 - math.sqrt((0.5 - (-0.5 / x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.22)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 - Float64(-0.5 / x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.22)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 1.0 - sqrt((0.5 - (-0.5 / x_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.22], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 53.1%

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

   if 1.21999999999999997 < 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{-1}{2} \cdot \frac{\color{blue}{1}}{x}} \]

      3. mult-flip N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 48.4

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

   8. Applied rewrites 48.4%

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

Alternative 10: 97.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* 0.125 (* x_m x_m)) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} \cdot \frac{1}{2}} \]

      13. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} \cdot \frac{1}{2}} \]

      14. pow2 N/A

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

      15. lower-fma.f64 75.3

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 53.1%

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

   if 1.55000000000000004 < 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 inf

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

   4. Applied rewrites 48.8%

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

Alternative 11: 73.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.2e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.2d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2e-77:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2e-77)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2e-77)
		tmp = 0.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2e-77], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000007e-77

   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

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

      1. sqrt-unprod N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot 2} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 28.4

         \[\leadsto 0 \]

   4. Applied rewrites 28.4%

      \[\leadsto \color{blue}{0} \]

   if 2.20000000000000007e-77 < 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 inf

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

   4. Applied rewrites 48.8%

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

Alternative 12: 28.4% accurate, 27.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

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

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

   1. sqrt-unprod N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot 2} \]

   2. metadata-eval N/A

      \[\leadsto 1 - \sqrt{1} \]

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 28.4

      \[\leadsto 0 \]

4. Applied rewrites 28.4%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))