Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Percentage Accurate: 65.3% → 99.6%

Time: 10.5s

Alternatives: 12

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+20}:\\ \;\;\;\;x - t_0\\ \mathbf{elif}\;y \leq 13800:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 \cdot \frac{1}{y} - \left(t_0 + \frac{x + -1}{{y}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -2.1e+20)
     (- x t_0)
     (if (<= y 13800.0)
       (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0)
       (+ x (- (* t_0 (/ 1.0 y)) (+ t_0 (/ (+ x -1.0) (pow y 3.0)))))))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -2.1e+20) {
		tmp = x - t_0;
	} else if (y <= 13800.0) {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	} else {
		tmp = x + ((t_0 * (1.0 / y)) - (t_0 + ((x + -1.0) / pow(y, 3.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -2.1e+20)
		tmp = Float64(x - t_0);
	elseif (y <= 13800.0)
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	else
		tmp = Float64(x + Float64(Float64(t_0 * Float64(1.0 / y)) - Float64(t_0 + Float64(Float64(x + -1.0) / (y ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -2.1e+20], N[(x - t$95$0), $MachinePrecision], If[LessEqual[y, 13800.0], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x + N[(N[(t$95$0 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 + N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1e20

   1. Initial program 31.1%

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

      1. associate-*l/ 67.5%

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

      2. +-commutative 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -2.1e20 < y < 13800

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-*l/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. neg-sub0 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]

      8. associate--r- 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]

      9. metadata-eval 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 13800 < y

   1. Initial program 30.9%

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

      1. associate-*l/ 53.2%

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

      2. +-commutative 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \frac{1}{{y}^{2}}\right)} \]

      2. associate-+r+ 100.0%

         \[\leadsto x + \left(\color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \frac{x}{{y}^{2}}\right)} - \frac{1}{{y}^{2}}\right) \]

      3. associate--l+ 100.0%

         \[\leadsto x + \color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)} \]

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

         \[\leadsto x + \left(\left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right) + \frac{\color{blue}{1 \cdot \left(x + -1\right)}}{{y}^{2}}\right) \]

      2. unpow2 100.0%

         \[\leadsto x + \left(\left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right) + \frac{1 \cdot \left(x + -1\right)}{\color{blue}{y \cdot y}}\right) \]

      3. times-frac 100.0%

         \[\leadsto x + \left(\left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right) + \color{blue}{\frac{1}{y} \cdot \frac{x + -1}{y}}\right) \]

   8. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+20}:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 13800:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{x + -1}{y} \cdot \frac{1}{y} - \left(\frac{x + -1}{y} + \frac{x + -1}{{y}^{3}}\right)\right)\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ t_1 := \frac{x + -1}{y + 1}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;1 + y \cdot t_1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;x + \left(\frac{x + -1}{y} \cdot \frac{1}{y} + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ y 1.0))) (t_1 (/ (+ x -1.0) (+ y 1.0))))
   (if (<= t_0 4e-8)
     (+ 1.0 (* y t_1))
     (if (<= t_0 1.0)
       (+
        x
        (+ (* (/ (+ x -1.0) y) (/ 1.0 y)) (+ (/ 1.0 y) (/ 1.0 (pow y 3.0)))))
       (fma t_1 y 1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double t_1 = (x + -1.0) / (y + 1.0);
	double tmp;
	if (t_0 <= 4e-8) {
		tmp = 1.0 + (y * t_1);
	} else if (t_0 <= 1.0) {
		tmp = x + ((((x + -1.0) / y) * (1.0 / y)) + ((1.0 / y) + (1.0 / pow(y, 3.0))));
	} else {
		tmp = fma(t_1, y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0))
	t_1 = Float64(Float64(x + -1.0) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 4e-8)
		tmp = Float64(1.0 + Float64(y * t_1));
	elseif (t_0 <= 1.0)
		tmp = Float64(x + Float64(Float64(Float64(Float64(x + -1.0) / y) * Float64(1.0 / y)) + Float64(Float64(1.0 / y) + Float64(1.0 / (y ^ 3.0)))));
	else
		tmp = fma(t_1, y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-8], N[(1.0 + N[(y * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(x + N[(N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / y), $MachinePrecision] + N[(1.0 / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * y + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 4.0000000000000001e-8

   1. Initial program 88.1%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 4.0000000000000001e-8 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1

   1. Initial program 6.7%

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

      1. associate-*l/ 6.5%

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

      2. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \frac{1}{{y}^{2}}\right)} \]

      2. associate-+r+ 100.0%

         \[\leadsto x + \left(\color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \frac{x}{{y}^{2}}\right)} - \frac{1}{{y}^{2}}\right) \]

      3. associate--l+ 100.0%

         \[\leadsto x + \color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)} \]

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

         \[\leadsto x + \left(\left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right) + \frac{\color{blue}{1 \cdot \left(x + -1\right)}}{{y}^{2}}\right) \]

      2. unpow2 100.0%

         \[\leadsto x + \left(\left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right) + \frac{1 \cdot \left(x + -1\right)}{\color{blue}{y \cdot y}}\right) \]

      3. times-frac 100.0%

         \[\leadsto x + \left(\left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right) + \color{blue}{\frac{1}{y} \cdot \frac{x + -1}{y}}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto x + \left(\left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right) + \color{blue}{\frac{1}{y} \cdot \frac{x + -1}{y}}\right) \]

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \left(\color{blue}{\left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)} + \frac{1}{y} \cdot \frac{x + -1}{y}\right) \]

   if 1 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 64.7%

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

      1. sub-neg 64.7%

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

      2. +-commutative 64.7%

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

      3. associate-*l/ 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. fma-def 99.7%

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

      6. distribute-frac-neg 99.7%

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

      7. neg-sub0 99.7%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]

      8. associate--r- 99.7%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]

      9. metadata-eval 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 1:\\ \;\;\;\;x + \left(\frac{x + -1}{y} \cdot \frac{1}{y} + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 220000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{x + -1}{{y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (+ x -1.0) y))))
   (if (<= y -2.1e+20)
     t_0
     (if (<= y 220000.0)
       (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0)
       (+ t_0 (/ (+ x -1.0) (pow y 2.0)))))))

C

double code(double x, double y) {
	double t_0 = x - ((x + -1.0) / y);
	double tmp;
	if (y <= -2.1e+20) {
		tmp = t_0;
	} else if (y <= 220000.0) {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	} else {
		tmp = t_0 + ((x + -1.0) / pow(y, 2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (y <= -2.1e+20)
		tmp = t_0;
	elseif (y <= 220000.0)
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	else
		tmp = Float64(t_0 + Float64(Float64(x + -1.0) / (y ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e+20], t$95$0, If[LessEqual[y, 220000.0], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(t$95$0 + N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1e20

   1. Initial program 31.1%

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

      1. associate-*l/ 67.5%

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

      2. +-commutative 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -2.1e20 < y < 2.2e5

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-*l/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. neg-sub0 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]

      8. associate--r- 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]

      9. metadata-eval 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 2.2e5 < y

   1. Initial program 30.9%

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

      1. associate-*l/ 53.2%

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

      2. +-commutative 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in y around -inf 99.7%

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

      1. associate-+r+ 99.7%

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

      2. associate--l+ 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. metadata-eval 99.7%

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

      7. div-sub 99.7%

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

      8. sub-neg 99.7%

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

      9. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+20}:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 220000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{x + -1}{y}\right) + \frac{x + -1}{{y}^{2}}\\ \end{array} \]

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+20} \lor \neg \left(y \leq 105000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e+20) (not (<= y 105000000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+20) || !(y <= 105000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (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 <= (-2.1d+20)) .or. (.not. (y <= 105000000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+20) || !(y <= 105000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.1e+20) or not (y <= 105000000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.1e+20) || !(y <= 105000000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.1e+20) || ~((y <= 105000000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.1e+20], N[Not[LessEqual[y, 105000000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1e20 or 1.05e8 < y

   1. Initial program 30.1%

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

      1. associate-*l/ 59.7%

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

      2. +-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -2.1e20 < y < 1.05e8

   1. Initial program 99.1%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+20} \lor \neg \left(y \leq 105000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 5: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16200000000 \lor \neg \left(y \leq 145000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -16200000000.0) (not (<= y 145000000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -16200000000.0) || !(y <= 145000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (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 <= (-16200000000.0d0)) .or. (.not. (y <= 145000000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -16200000000.0) || !(y <= 145000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -16200000000.0) or not (y <= 145000000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -16200000000.0) || !(y <= 145000000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -16200000000.0) || ~((y <= 145000000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -16200000000.0], N[Not[LessEqual[y, 145000000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.62e10 or 1.45e8 < y

   1. Initial program 30.0%

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

      1. associate-*l/ 60.0%

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

      2. +-commutative 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -1.62e10 < y < 1.45e8

   1. Initial program 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16200000000 \lor \neg \left(y \leq 145000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 6: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-24} \lor \neg \left(y \leq 1.9 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e-24) (not (<= y 1.9e-7))) (* x (/ y (+ y 1.0))) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e-24) || !(y <= 1.9e-7)) {
		tmp = x * (y / (y + 1.0));
	} else {
		tmp = 1.0 - 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 <= (-7d-24)) .or. (.not. (y <= 1.9d-7))) then
        tmp = x * (y / (y + 1.0d0))
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7e-24) || !(y <= 1.9e-7)) {
		tmp = x * (y / (y + 1.0));
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e-24) or not (y <= 1.9e-7):
		tmp = x * (y / (y + 1.0))
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e-24) || !(y <= 1.9e-7))
		tmp = Float64(x * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7e-24) || ~((y <= 1.9e-7)))
		tmp = x * (y / (y + 1.0));
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e-24], N[Not[LessEqual[y, 1.9e-7]], $MachinePrecision]], N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999993e-24 or 1.90000000000000007e-7 < y

   1. Initial program 35.5%

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

      1. associate-*l/ 63.0%

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

      2. +-commutative 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in x around inf 50.0%

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

      1. associate-*r/ 77.6%

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

      2. *-commutative 77.6%

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

   6. Simplified 77.6%

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

   if -6.9999999999999993e-24 < y < 1.90000000000000007e-7

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.4%

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

   5. Taylor expanded in y around 0 74.4%

      \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 74.4%

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

      2. unsub-neg 74.4%

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

   7. Simplified 74.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-24} \lor \neg \left(y \leq 1.9 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 7: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-16} \lor \neg \left(y \leq 1.46 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.3e-16) (not (<= y 1.46e-5)))
   (* x (/ y (+ y 1.0)))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e-16) || !(y <= 1.46e-5)) {
		tmp = x * (y / (y + 1.0));
	} else {
		tmp = 1.0 + (y * (x + -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 <= (-3.3d-16)) .or. (.not. (y <= 1.46d-5))) then
        tmp = x * (y / (y + 1.0d0))
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e-16) || !(y <= 1.46e-5)) {
		tmp = x * (y / (y + 1.0));
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.3e-16) or not (y <= 1.46e-5):
		tmp = x * (y / (y + 1.0))
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.3e-16) || !(y <= 1.46e-5))
		tmp = Float64(x * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.3e-16) || ~((y <= 1.46e-5)))
		tmp = x * (y / (y + 1.0));
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.3e-16], N[Not[LessEqual[y, 1.46e-5]], $MachinePrecision]], N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.29999999999999988e-16 or 1.46000000000000008e-5 < y

   1. Initial program 35.0%

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

      1. associate-*l/ 62.7%

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

      2. +-commutative 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in x around inf 49.6%

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

      1. associate-*r/ 77.5%

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

      2. *-commutative 77.5%

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

   6. Simplified 77.5%

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

   if -3.29999999999999988e-16 < y < 1.46000000000000008e-5

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-16} \lor \neg \left(y \leq 1.46 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 8: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * (x + -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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 31.9%

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

      1. associate-*l/ 61.0%

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

      2. +-commutative 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in y around inf 98.8%

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

      1. associate--l+ 98.8%

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

      2. div-sub 98.8%

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

      3. sub-neg 98.8%

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

      4. +-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. distribute-neg-in 98.8%

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

      7. distribute-neg-frac 98.8%

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

      8. metadata-eval 98.8%

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

      9. sub-neg 98.8%

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

      10. unsub-neg 98.8%

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

      11. sub-neg 98.8%

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

      12. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.7%

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

4. Final simplification 98.3%

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

Alternative 9: 74.3% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (- x (/ x y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x - (x / y);
	} else {
		tmp = 1.0 - 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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = x - (x / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x - (x / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = x - (x / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 31.9%

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

      1. associate-*l/ 61.0%

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

      2. +-commutative 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in x around inf 47.2%

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

      1. associate-*r/ 76.4%

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

      2. *-commutative 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in y around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. unsub-neg 76.2%

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

   9. Simplified 76.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 70.6%

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

   5. Taylor expanded in y around 0 70.6%

      \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 70.6%

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

      2. unsub-neg 70.6%

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

   7. Simplified 70.6%

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

4. Final simplification 73.3%

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

Alternative 10: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.000235:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.000235) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.000235) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.000235d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.000235) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.000235:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.000235)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.000235)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.000235], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.34999999999999993e-4 < y

   1. Initial program 33.0%

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

      1. associate-*l/ 61.5%

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

      2. +-commutative 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in y around inf 74.7%

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

   if -1 < y < 2.34999999999999993e-4

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 71.6%

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

   5. Taylor expanded in y around 0 71.6%

      \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 71.6%

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

      2. unsub-neg 71.6%

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

   7. Simplified 71.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.000235:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 73.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.000235:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.000235) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.000235) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.000235d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.000235) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.000235:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.000235)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.000235)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.000235], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.34999999999999993e-4 < y

   1. Initial program 33.0%

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

      1. associate-*l/ 61.5%

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

      2. +-commutative 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in y around inf 74.7%

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

   if -1 < y < 2.34999999999999993e-4

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 71.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.000235:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 38.0% accurate, 11.0× 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 66.8%

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

   1. associate-*l/ 80.9%

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

   2. +-commutative 80.9%

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

3. Simplified 80.9%

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

4. Taylor expanded in y around 0 37.7%

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

5. Final simplification 37.7%

   \[\leadsto 1 \]

Developer target: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 y) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_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 = (1.0d0 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023305 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))