ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.5%

Time: 6.4s

Alternatives: 5

Speedup: 1.1×

Specification

\[0.999 \leq x \land x \leq 1.001\]

Math

\[\begin{array}{l} \\ \frac{10}{1 - x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 10.0 (- 1.0 (* x x))))

C

double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 10.0d0 / (1.0d0 - (x * x))
end function

Java

public static double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

Python

def code(x):
	return 10.0 / (1.0 - (x * x))

Julia

function code(x)
	return Float64(10.0 / Float64(1.0 - Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 10.0 / (1.0 - (x * x));
end

Wolfram

code[x_] := N[(10.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{10}{1 - x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 10.0 (- 1.0 (* x x))))

C

double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 10.0d0 / (1.0d0 - (x * x))
end function

Java

public static double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

Python

def code(x):
	return 10.0 / (1.0 - (x * x))

Julia

function code(x)
	return Float64(10.0 / Float64(1.0 - Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 10.0 / (1.0 - (x * x));
end

Wolfram

code[x_] := N[(10.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{4}\\ \frac{-10 \cdot t\_0}{\mathsf{fma}\left(t\_0, \left(x - 1\right) \cdot x, t\_0 \cdot \left(x - 1\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x 4.0))))
   (/ (* -10.0 t_0) (fma t_0 (* (- x 1.0) x) (* t_0 (- x 1.0))))))

C

double code(double x) {
	double t_0 = 1.0 - pow(x, 4.0);
	return (-10.0 * t_0) / fma(t_0, ((x - 1.0) * x), (t_0 * (x - 1.0)));
}

Julia

function code(x)
	t_0 = Float64(1.0 - (x ^ 4.0))
	return Float64(Float64(-10.0 * t_0) / fma(t_0, Float64(Float64(x - 1.0) * x), Float64(t_0 * Float64(x - 1.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(-10.0 * t$95$0), $MachinePrecision] / N[(t$95$0 * N[(N[(x - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(t$95$0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{10}{1 - x \cdot x}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{10}{\color{blue}{1 - x \cdot x}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{10}{1 - \color{blue}{x \cdot x}} \]

   4. rem-square-sqrt N/A

      \[\leadsto \frac{10}{1 - x \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

   5. sqrt-prod N/A

      \[\leadsto \frac{10}{1 - x \cdot \color{blue}{\sqrt{x \cdot x}}} \]

   6. sqr-neg-rev N/A

      \[\leadsto \frac{10}{1 - x \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}} \]

   7. pow2 N/A

      \[\leadsto \frac{10}{1 - x \cdot \sqrt{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{2}}}} \]

   8. sqrt-pow1 N/A

      \[\leadsto \frac{10}{1 - x \cdot \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{2}{2}\right)}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{10}{1 - x \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\color{blue}{1}}} \]

   10. unpow1 N/A

       \[\leadsto \frac{10}{1 - x \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}} \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \frac{10}{1 - \color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   12. sqr-abs-rev N/A

       \[\leadsto \frac{10}{1 - \left(\mathsf{neg}\left(\color{blue}{\left|x\right| \cdot \left|x\right|}\right)\right)} \]

   13. sqr-neg-rev N/A

       \[\leadsto \frac{10}{1 - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left|x\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}\right)\right)} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \frac{10}{1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}} \]

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

       \[\leadsto \frac{10}{\color{blue}{1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}} \]

   16. sqr-neg-rev N/A

       \[\leadsto \frac{10}{1 + \color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   17. sqr-abs-rev N/A

       \[\leadsto \frac{10}{1 + \color{blue}{x \cdot x}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{10}{1 + \color{blue}{x \cdot x}} \]

   19. frac-2neg N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(10\right)}{\mathsf{neg}\left(\left(1 + x \cdot x\right)\right)}} \]

   20. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(10\right)}{\mathsf{neg}\left(\left(1 + x \cdot x\right)\right)}} \]

   21. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{-10}}{\mathsf{neg}\left(\left(1 + x \cdot x\right)\right)} \]

   22. +-commutative N/A

       \[\leadsto \frac{-10}{\mathsf{neg}\left(\color{blue}{\left(x \cdot x + 1\right)}\right)} \]

   23. distribute-neg-in N/A

       \[\leadsto \frac{-10}{\color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{-10}{\color{blue}{x \cdot x + -1}} \]

   2. difference-of-sqr--1 N/A

      \[\leadsto \frac{-10}{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]

   3. difference-of-sqr-1-rev N/A

      \[\leadsto \frac{-10}{\color{blue}{x \cdot x - 1}} \]

   4. pow2 N/A

      \[\leadsto \frac{-10}{\color{blue}{{x}^{2}} - 1} \]

   5. pow-to-exp N/A

      \[\leadsto \frac{-10}{\color{blue}{e^{\log x \cdot 2}} - 1} \]

   6. rem-log-exp N/A

      \[\leadsto \frac{-10}{e^{\color{blue}{\log \left(e^{\log x \cdot 2}\right)}} - 1} \]

   7. pow-to-exp N/A

      \[\leadsto \frac{-10}{e^{\log \color{blue}{\left({x}^{2}\right)}} - 1} \]

   8. pow2 N/A

      \[\leadsto \frac{-10}{e^{\log \color{blue}{\left(x \cdot x\right)}} - 1} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-10}{e^{\log \color{blue}{\left(x \cdot x\right)}} - 1} \]

   10. lower-expm1.f64 N/A

       \[\leadsto \frac{-10}{\color{blue}{\mathsf{expm1}\left(\log \left(x \cdot x\right)\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{-10}{\mathsf{expm1}\left(\log \color{blue}{\left(x \cdot x\right)}\right)} \]

   12. pow2 N/A

       \[\leadsto \frac{-10}{\mathsf{expm1}\left(\log \color{blue}{\left({x}^{2}\right)}\right)} \]

   13. log-pow N/A

       \[\leadsto \frac{-10}{\mathsf{expm1}\left(\color{blue}{2 \cdot \log x}\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{-10}{\mathsf{expm1}\left(\color{blue}{2 \cdot \log x}\right)} \]

   15. lower-log.f64 99.4

       \[\leadsto \frac{-10}{\mathsf{expm1}\left(2 \cdot \color{blue}{\log x}\right)} \]

6. Applied rewrites 99.4%

   \[\leadsto \frac{-10}{\color{blue}{\mathsf{expm1}\left(2 \cdot \log x\right)}} \]

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \mathsf{fma}\left(x, x, -1\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\color{blue}{\left(1 - {x}^{4}\right) \cdot \mathsf{fma}\left(x, x, -1\right)}} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \color{blue}{\left(x \cdot x + -1\right)}} \]

   3. difference-of-sqr--1 N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \color{blue}{\left(\left(x + 1\right) \cdot \left(x - 1\right)\right)}} \]

   4. rem-square-sqrt N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) \cdot \left(x - 1\right)\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\left(\color{blue}{\sqrt{x}} \cdot \sqrt{x} + 1\right) \cdot \left(x - 1\right)\right)} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{x}} + 1\right) \cdot \left(x - 1\right)\right)} \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)} \cdot \left(x - 1\right)\right)} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right) \cdot \color{blue}{\left(x - 1\right)}\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \color{blue}{\left(\left(x - 1\right) \cdot \mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)\right)}} \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\left(x - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x} + 1\right)}\right)} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\left(x - 1\right) \cdot \left(\color{blue}{\sqrt{x}} \cdot \sqrt{x} + 1\right)\right)} \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\left(x - 1\right) \cdot \left(\sqrt{x} \cdot \color{blue}{\sqrt{x}} + 1\right)\right)} \]

   13. rem-square-sqrt N/A

       \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \left(\left(x - 1\right) \cdot \left(\color{blue}{x} + 1\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\left(1 - {x}^{4}\right) \cdot \color{blue}{\left(\left(x - 1\right) \cdot x + \left(x - 1\right) \cdot 1\right)}} \]

   15. distribute-lft-in N/A

       \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\color{blue}{\left(1 - {x}^{4}\right) \cdot \left(\left(x - 1\right) \cdot x\right) + \left(1 - {x}^{4}\right) \cdot \left(\left(x - 1\right) \cdot 1\right)}} \]

10. Applied rewrites 99.5%

    \[\leadsto \frac{-10 \cdot \left(1 - {x}^{4}\right)}{\color{blue}{\mathsf{fma}\left(1 - {x}^{4}, \left(x - 1\right) \cdot x, \left(1 - {x}^{4}\right) \cdot \left(x - 1\right)\right)}} \]
11. Add Preprocessing

Alternative 2: 13.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{10}{1 - x \cdot x} \leq -5000:\\ \;\;\;\;\left(-10 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot 10\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 10.0 (- 1.0 (* x x))) -5000.0)
   (* (* -10.0 x) x)
   (* (fma x x 1.0) 10.0)))

C

double code(double x) {
	double tmp;
	if ((10.0 / (1.0 - (x * x))) <= -5000.0) {
		tmp = (-10.0 * x) * x;
	} else {
		tmp = fma(x, x, 1.0) * 10.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(10.0 / Float64(1.0 - Float64(x * x))) <= -5000.0)
		tmp = Float64(Float64(-10.0 * x) * x);
	else
		tmp = Float64(fma(x, x, 1.0) * 10.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(10.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5000.0], N[(N[(-10.0 * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * 10.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 10 binary64) (-.f64 #s(literal 1 binary64) (*.f64 x x))) < -5e3

   1. Initial program 86.7%

      \[\frac{10}{1 - x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{10 + 10 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{10 \cdot {x}^{2} + 10} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot 10} + 10 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 10, 10\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 10, 10\right) \]

      5. lower-*.f64 1.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 10, 10\right) \]

   5. Applied rewrites 1.5%

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

   7. Applied rewrites 1.5%

      \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \color{blue}{10} \]

   9. Applied rewrites 11.9%

      \[\leadsto \mathsf{fma}\left(-x, x, 1\right) \cdot 10 \]

   10. Taylor expanded in x around inf

       \[\leadsto -10 \cdot \color{blue}{{x}^{2}} \]
   11. Step-by-step derivation

   12. Applied rewrites 13.5%

       \[\leadsto \left(-10 \cdot x\right) \cdot \color{blue}{x} \]

   if -5e3 < (/.f64 #s(literal 10 binary64) (-.f64 #s(literal 1 binary64) (*.f64 x x)))

   1. Initial program 88.3%

      \[\frac{10}{1 - x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{10 + 10 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{10 \cdot {x}^{2} + 10} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot 10} + 10 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 10, 10\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 10, 10\right) \]

      5. lower-*.f64 13.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 10, 10\right) \]

   5. Applied rewrites 13.7%

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

   7. Applied rewrites 13.7%

      \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \color{blue}{10} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 13.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\left(-10 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;10\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -1e-6) (* (* -10.0 x) x) 10.0))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -1e-6) {
		tmp = (-10.0 * x) * x;
	} else {
		tmp = 10.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 - (x * x)) <= (-1d-6)) then
        tmp = ((-10.0d0) * x) * x
    else
        tmp = 10.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -1e-6) {
		tmp = (-10.0 * x) * x;
	} else {
		tmp = 10.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - (x * x)) <= -1e-6:
		tmp = (-10.0 * x) * x
	else:
		tmp = 10.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -1e-6)
		tmp = Float64(Float64(-10.0 * x) * x);
	else
		tmp = 10.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - (x * x)) <= -1e-6)
		tmp = (-10.0 * x) * x;
	else
		tmp = 10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -1e-6], N[(N[(-10.0 * x), $MachinePrecision] * x), $MachinePrecision], 10.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -9.99999999999999955e-7

   1. Initial program 86.7%

      \[\frac{10}{1 - x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{10 + 10 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{10 \cdot {x}^{2} + 10} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot 10} + 10 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 10, 10\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 10, 10\right) \]

      5. lower-*.f64 1.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 10, 10\right) \]

   5. Applied rewrites 1.5%

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

   7. Applied rewrites 1.5%

      \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \color{blue}{10} \]

   9. Applied rewrites 11.9%

      \[\leadsto \mathsf{fma}\left(-x, x, 1\right) \cdot 10 \]

   10. Taylor expanded in x around inf

       \[\leadsto -10 \cdot \color{blue}{{x}^{2}} \]
   11. Step-by-step derivation

   12. Applied rewrites 13.5%

       \[\leadsto \left(-10 \cdot x\right) \cdot \color{blue}{x} \]

   if -9.99999999999999955e-7 < (-.f64 #s(literal 1 binary64) (*.f64 x x))

   1. Initial program 88.3%

      \[\frac{10}{1 - x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{10} \]
   4. Step-by-step derivation

   5. Applied rewrites 13.5%

      \[\leadsto \color{blue}{10} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -10.0 (fma x x -1.0)))

C

double code(double x) {
	return -10.0 / fma(x, x, -1.0);
}

Julia

function code(x)
	return Float64(-10.0 / fma(x, x, -1.0))
end

Wolfram

code[x_] := N[(-10.0 / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{10}{1 - x \cdot x}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{10}{\color{blue}{1 - x \cdot x}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{10}{1 - \color{blue}{x \cdot x}} \]

   4. rem-square-sqrt N/A

      \[\leadsto \frac{10}{1 - x \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

   5. sqrt-prod N/A

      \[\leadsto \frac{10}{1 - x \cdot \color{blue}{\sqrt{x \cdot x}}} \]

   6. sqr-neg-rev N/A

      \[\leadsto \frac{10}{1 - x \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}} \]

   7. pow2 N/A

      \[\leadsto \frac{10}{1 - x \cdot \sqrt{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{2}}}} \]

   8. sqrt-pow1 N/A

      \[\leadsto \frac{10}{1 - x \cdot \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{2}{2}\right)}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{10}{1 - x \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\color{blue}{1}}} \]

   10. unpow1 N/A

       \[\leadsto \frac{10}{1 - x \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}} \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \frac{10}{1 - \color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}} \]

   12. sqr-abs-rev N/A

       \[\leadsto \frac{10}{1 - \left(\mathsf{neg}\left(\color{blue}{\left|x\right| \cdot \left|x\right|}\right)\right)} \]

   13. sqr-neg-rev N/A

       \[\leadsto \frac{10}{1 - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left|x\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}\right)\right)} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \frac{10}{1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}} \]

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

       \[\leadsto \frac{10}{\color{blue}{1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}} \]

   16. sqr-neg-rev N/A

       \[\leadsto \frac{10}{1 + \color{blue}{\left|x\right| \cdot \left|x\right|}} \]

   17. sqr-abs-rev N/A

       \[\leadsto \frac{10}{1 + \color{blue}{x \cdot x}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{10}{1 + \color{blue}{x \cdot x}} \]

   19. frac-2neg N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(10\right)}{\mathsf{neg}\left(\left(1 + x \cdot x\right)\right)}} \]

   20. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(10\right)}{\mathsf{neg}\left(\left(1 + x \cdot x\right)\right)}} \]

   21. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{-10}}{\mathsf{neg}\left(\left(1 + x \cdot x\right)\right)} \]

   22. +-commutative N/A

       \[\leadsto \frac{-10}{\mathsf{neg}\left(\color{blue}{\left(x \cdot x + 1\right)}\right)} \]

   23. distribute-neg-in N/A

       \[\leadsto \frac{-10}{\color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
5. Add Preprocessing

Alternative 5: 9.5% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ 10 \end{array} \]

FPCore

(FPCore (x) :precision binary64 10.0)

C

double code(double x) {
	return 10.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 10.0d0
end function

Java

public static double code(double x) {
	return 10.0;
}

Python

def code(x):
	return 10.0

Julia

function code(x)
	return 10.0
end

MATLAB

function tmp = code(x)
	tmp = 10.0;
end

Wolfram

code[x_] := 10.0

Derivation

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{10} \]
4. Step-by-step derivation

5. Applied rewrites 9.4%

   \[\leadsto \color{blue}{10} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, B"
  :precision binary64
  :pre (and (<= 0.999 x) (<= x 1.001))
  (/ 10.0 (- 1.0 (* x x))))