Kahan p9 Example

Percentage Accurate: 68.2% → 99.9%

Time: 11.1s

Alternatives: 10

Speedup: 0.5×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

Wolfram

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

Initial Program: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Step-by-step derivation

   1. add-sqr-sqrt 70.3%

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

   2. times-frac 70.2%

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

   3. hypot-def 70.3%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. clear-num 99.9%

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

   3. frac-times 100.0%

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

   4. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (* (/ (- x y) (hypot x y)) (/ (+ x y) (hypot x y))))

C

double code(double x, double y) {
	return ((x - y) / hypot(x, y)) * ((x + y) / hypot(x, y));
}

Java

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

Python

def code(x, y):
	return ((x - y) / math.hypot(x, y)) * ((x + y) / math.hypot(x, y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) / hypot(x, y)) * Float64(Float64(x + y) / hypot(x, y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) / hypot(x, y)) * ((x + y) / hypot(x, y));
end

Wolfram

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

Derivation

1. Initial program 70.3%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Step-by-step derivation

   1. add-sqr-sqrt 70.3%

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

   2. times-frac 70.2%

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

   3. hypot-def 70.3%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]

Alternative 3: 100.0% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (+ x y) (* (hypot x y) (/ (hypot x y) (- x y)))))

C

double code(double x, double y) {
	return (x + y) / (hypot(x, y) * (hypot(x, y) / (x - y)));
}

Java

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

Python

def code(x, y):
	return (x + y) / (math.hypot(x, y) * (math.hypot(x, y) / (x - y)))

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(hypot(x, y) * Float64(hypot(x, y) / Float64(x - y))))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / (hypot(x, y) * (hypot(x, y) / (x - y)));
end

Wolfram

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

Derivation

1. Initial program 70.3%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Step-by-step derivation

   1. add-sqr-sqrt 70.3%

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

   2. times-frac 70.2%

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

   3. hypot-def 70.3%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. frac-times 99.9%

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

   3. *-un-lft-identity 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

   \[\leadsto \frac{x + y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x - y}} \]

Alternative 4: 100.0% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (+ x y) (/ (hypot x y) (/ (- x y) (hypot x y)))))

C

double code(double x, double y) {
	return (x + y) / (hypot(x, y) / ((x - y) / hypot(x, y)));
}

Java

public static double code(double x, double y) {
	return (x + y) / (Math.hypot(x, y) / ((x - y) / Math.hypot(x, y)));
}

Python

def code(x, y):
	return (x + y) / (math.hypot(x, y) / ((x - y) / math.hypot(x, y)))

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(hypot(x, y) / Float64(Float64(x - y) / hypot(x, y))))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / (hypot(x, y) / ((x - y) / hypot(x, y)));
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Step-by-step derivation

   1. add-sqr-sqrt 70.3%

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

   2. times-frac 70.2%

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

   3. hypot-def 70.3%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. frac-times 99.9%

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

   3. *-un-lft-identity 99.9%

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

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{x + y}{\frac{\mathsf{hypot}\left(x, y\right)}{x - y} \cdot \mathsf{hypot}\left(x, y\right)}} \]
6. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. clear-num 99.9%

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

   3. un-div-inv 100.0%

      \[\leadsto \frac{x + y}{\color{blue}{\frac{\mathsf{hypot}\left(x, y\right)}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}}} \]

7. Applied egg-rr 100.0%

   \[\leadsto \frac{x + y}{\color{blue}{\frac{\mathsf{hypot}\left(x, y\right)}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}}} \]

8. Final simplification 100.0%

   \[\leadsto \frac{x + y}{\frac{\mathsf{hypot}\left(x, y\right)}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}} \]

Alternative 5: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (+ (* 2.0 (* (/ x y) (/ x y))) -1.0))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (2.0d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(2.0 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. associate-*r/ 3.1%

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

      4. +-commutative 3.1%

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

      5. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 65.8%

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

      1. fma-neg 65.8%

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

      2. unpow2 65.8%

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

      3. unpow2 65.8%

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

      4. times-frac 87.5%

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

      5. metadata-eval 87.5%

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

   6. Simplified 87.5%

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

      1. fma-udef 87.5%

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

      2. pow2 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. unpow2 87.5%

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

   10. Applied egg-rr 87.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]

Alternative 6: 42.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-138}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e-138) (+ 1.0 (* -2.0 (* (/ y x) (/ y x)))) (/ (- x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-138) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8d-138) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e-138) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-138:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	else:
		tmp = (x - y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-138)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e-138)
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-138], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.00000000000000054e-138

   1. Initial program 63.6%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 63.6%

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

      2. +-commutative 63.6%

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

      3. associate-*r/ 63.3%

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

      4. +-commutative 63.3%

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

      5. fma-def 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in y around 0 26.6%

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

      1. unpow2 26.6%

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

      2. unpow2 26.6%

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

   6. Simplified 26.6%

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

      1. frac-times 31.5%

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

   8. Applied egg-rr 31.5%

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

   if 8.00000000000000054e-138 < y

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

      7. sub-neg 99.9%

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

      8. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 71.8%

      \[\leadsto \frac{x - y}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-138}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 7: 43.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-142}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.32e-142)
   (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))
   (+ (* 2.0 (* (/ x y) (/ x y))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.32e-142) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.32d-142) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else
        tmp = (2.0d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.32e-142) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.32e-142:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	else:
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.32e-142)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.32e-142)
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	else
		tmp = (2.0 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.32e-142], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.32e-142

   1. Initial program 63.6%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 63.6%

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

      2. +-commutative 63.6%

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

      3. associate-*r/ 63.3%

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

      4. +-commutative 63.3%

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

      5. fma-def 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in y around 0 26.6%

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

      1. unpow2 26.6%

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

      2. unpow2 26.6%

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

   6. Simplified 26.6%

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

      1. frac-times 31.5%

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

   8. Applied egg-rr 31.5%

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

   if 1.32e-142 < y

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r/ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 72.9%

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

      1. fma-neg 72.9%

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

      2. unpow2 72.9%

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

      3. unpow2 72.9%

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

      4. times-frac 72.9%

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

      5. metadata-eval 72.9%

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

   6. Simplified 72.9%

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

      1. fma-udef 72.9%

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

      2. pow2 72.9%

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

   8. Applied egg-rr 72.9%

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

      1. unpow2 72.9%

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

   10. Applied egg-rr 72.9%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{-142}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]

Alternative 8: 41.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-141}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5e-141) 1.0 (/ (- x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e-141) {
		tmp = 1.0;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d-141) then
        tmp = 1.0d0
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-141) {
		tmp = 1.0;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e-141:
		tmp = 1.0
	else:
		tmp = (x - y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e-141)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-141)
		tmp = 1.0;
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e-141], 1.0, N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e-141

   1. Initial program 63.6%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 63.6%

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

      2. +-commutative 63.6%

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

      3. associate-*r/ 63.3%

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

      4. +-commutative 63.3%

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

      5. fma-def 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in x around inf 29.5%

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

   if 4.9999999999999999e-141 < y

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

      7. sub-neg 99.9%

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

      8. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 71.8%

      \[\leadsto \frac{x - y}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-141}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 9: 41.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-141}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.6e-141) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-141) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.6d-141) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-141) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.6e-141:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.6e-141)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.6e-141)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.6e-141], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999999e-141

   1. Initial program 63.6%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 63.6%

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

      2. +-commutative 63.6%

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

      3. associate-*r/ 63.3%

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

      4. +-commutative 63.3%

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

      5. fma-def 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in x around inf 29.5%

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

   if 4.5999999999999999e-141 < y

   1. Initial program 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r/ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 70.8%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-141}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 67.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 70.3%

   \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
2. Step-by-step derivation

   1. +-commutative 70.3%

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

   2. +-commutative 70.3%

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

   3. associate-*r/ 70.0%

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

   4. +-commutative 70.0%

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

   5. fma-def 70.0%

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

3. Simplified 70.0%

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

4. Taylor expanded in x around 0 70.6%

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

5. Final simplification 70.6%

   \[\leadsto -1 \]

Developer target: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;0.5 < t_0 \land t_0 < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (and (< 0.5 t_0) (< t_0 2.0))
     (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))
     (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = fabs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((x / y))
	tmp = 0
	if (0.5 < t_0) and (t_0 < 2.0):
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))))
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(2.0 / Float64(1.0 + Float64(Float64(x / y) * Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((x / y));
	tmp = 0.0;
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[And[Less[0.5, t$95$0], Less[t$95$0, 2.0]], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / N[(1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))