Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Percentage Accurate: 84.3% → 97.0%

Time: 10.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y - z\right)}{y} \leq -1 \cdot 10^{+72}:\\ \;\;\;\;x\_m - \frac{x\_m \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{y}{y - z}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (- y z)) y) -1e+72)
    (- x_m (/ (* x_m z) y))
    (/ x_m (/ y (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (y - z)) / y) <= -1e+72) {
		tmp = x_m - ((x_m * z) / y);
	} else {
		tmp = x_m / (y / (y - z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (y - z)) / y) <= (-1d+72)) then
        tmp = x_m - ((x_m * z) / y)
    else
        tmp = x_m / (y / (y - z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (y - z)) / y) <= -1e+72) {
		tmp = x_m - ((x_m * z) / y);
	} else {
		tmp = x_m / (y / (y - z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (y - z)) / y) <= -1e+72:
		tmp = x_m - ((x_m * z) / y)
	else:
		tmp = x_m / (y / (y - z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(y - z)) / y) <= -1e+72)
		tmp = Float64(x_m - Float64(Float64(x_m * z) / y));
	else
		tmp = Float64(x_m / Float64(y / Float64(y - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((x_m * (y - z)) / y) <= -1e+72)
		tmp = x_m - ((x_m * z) / y);
	else
		tmp = x_m / (y / (y - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -1e+72], N[(x$95$m - N[(N[(x$95$m * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -9.99999999999999944e71

   1. Initial program 91.2%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -9.99999999999999944e71 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

Alternative 2: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y - z\right)}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-146}:\\ \;\;\;\;\frac{x\_m \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+223}:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{-z}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (- y z)) y)))
   (*
    x_s
    (if (<= t_0 -5e-146)
      (/ (* x_m (- z)) y)
      (if (<= t_0 5e+223) (* x_m 1.0) (* x_m (/ (- z) y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y - z)) / y;
	double tmp;
	if (t_0 <= -5e-146) {
		tmp = (x_m * -z) / y;
	} else if (t_0 <= 5e+223) {
		tmp = x_m * 1.0;
	} else {
		tmp = x_m * (-z / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (y - z)) / y
    if (t_0 <= (-5d-146)) then
        tmp = (x_m * -z) / y
    else if (t_0 <= 5d+223) then
        tmp = x_m * 1.0d0
    else
        tmp = x_m * (-z / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y - z)) / y;
	double tmp;
	if (t_0 <= -5e-146) {
		tmp = (x_m * -z) / y;
	} else if (t_0 <= 5e+223) {
		tmp = x_m * 1.0;
	} else {
		tmp = x_m * (-z / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (y - z)) / y
	tmp = 0
	if t_0 <= -5e-146:
		tmp = (x_m * -z) / y
	elif t_0 <= 5e+223:
		tmp = x_m * 1.0
	else:
		tmp = x_m * (-z / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(y - z)) / y)
	tmp = 0.0
	if (t_0 <= -5e-146)
		tmp = Float64(Float64(x_m * Float64(-z)) / y);
	elseif (t_0 <= 5e+223)
		tmp = Float64(x_m * 1.0);
	else
		tmp = Float64(x_m * Float64(Float64(-z) / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (y - z)) / y;
	tmp = 0.0;
	if (t_0 <= -5e-146)
		tmp = (x_m * -z) / y;
	elseif (t_0 <= 5e+223)
		tmp = x_m * 1.0;
	else
		tmp = x_m * (-z / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -5e-146], N[(N[(x$95$m * (-z)), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 5e+223], N[(x$95$m * 1.0), $MachinePrecision], N[(x$95$m * N[((-z) / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -4.99999999999999957e-146

   1. Initial program 93.9%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      2. lower-neg.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if -4.99999999999999957e-146 < (/.f64 (*.f64 x (-.f64 y z)) y) < 4.99999999999999985e223

   1. Initial program 91.8%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      2. lower-neg.f64 33.6

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

   5. Applied rewrites 33.6%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{neg}\left(z\right)\right)}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\mathsf{neg}\left(z\right)}{y}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      6. lower-/.f64 34.6

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

   7. Applied rewrites 34.6%

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

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 63.6%

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

   if 4.99999999999999985e223 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 67.5%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      2. lower-neg.f64 65.4

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

   5. Applied rewrites 65.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{neg}\left(z\right)\right)}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\mathsf{neg}\left(z\right)}{y}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      6. lower-/.f64 65.9

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

   7. Applied rewrites 65.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -5 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 5 \cdot 10^{+223}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y - z\right)}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-z\right) \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+223}:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{-z}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (- y z)) y)))
   (*
    x_s
    (if (<= t_0 0.0)
      (* (- z) (/ x_m y))
      (if (<= t_0 5e+223) (* x_m 1.0) (* x_m (/ (- z) y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y - z)) / y;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -z * (x_m / y);
	} else if (t_0 <= 5e+223) {
		tmp = x_m * 1.0;
	} else {
		tmp = x_m * (-z / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (y - z)) / y
    if (t_0 <= 0.0d0) then
        tmp = -z * (x_m / y)
    else if (t_0 <= 5d+223) then
        tmp = x_m * 1.0d0
    else
        tmp = x_m * (-z / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y - z)) / y;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -z * (x_m / y);
	} else if (t_0 <= 5e+223) {
		tmp = x_m * 1.0;
	} else {
		tmp = x_m * (-z / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (y - z)) / y
	tmp = 0
	if t_0 <= 0.0:
		tmp = -z * (x_m / y)
	elif t_0 <= 5e+223:
		tmp = x_m * 1.0
	else:
		tmp = x_m * (-z / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(y - z)) / y)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-z) * Float64(x_m / y));
	elseif (t_0 <= 5e+223)
		tmp = Float64(x_m * 1.0);
	else
		tmp = Float64(x_m * Float64(Float64(-z) / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (y - z)) / y;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -z * (x_m / y);
	elseif (t_0 <= 5e+223)
		tmp = x_m * 1.0;
	else
		tmp = x_m * (-z / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0], N[((-z) * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+223], N[(x$95$m * 1.0), $MachinePrecision], N[(x$95$m * N[((-z) / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -0.0

   1. Initial program 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      2. lower-neg.f64 51.3

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

   7. Applied rewrites 51.3%

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

   if -0.0 < (/.f64 (*.f64 x (-.f64 y z)) y) < 4.99999999999999985e223

   1. Initial program 99.7%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      2. lower-neg.f64 40.1

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

   5. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{neg}\left(z\right)\right)}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\mathsf{neg}\left(z\right)}{y}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      6. lower-/.f64 35.4

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 61.1%

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

   if 4.99999999999999985e223 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 67.5%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      2. lower-neg.f64 65.4

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

   5. Applied rewrites 65.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{neg}\left(z\right)\right)}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\mathsf{neg}\left(z\right)}{y}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      6. lower-/.f64 65.9

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

   7. Applied rewrites 65.9%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq 0:\\ \;\;\;\;\left(-z\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 5 \cdot 10^{+223}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y - z\right)}{y} \leq -2 \cdot 10^{+79}:\\ \;\;\;\;x\_m - \frac{x\_m \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{-y}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (- y z)) y) -2e+79)
    (- x_m (/ (* x_m z) y))
    (fma (/ z (- y)) x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (y - z)) / y) <= -2e+79) {
		tmp = x_m - ((x_m * z) / y);
	} else {
		tmp = fma((z / -y), x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(y - z)) / y) <= -2e+79)
		tmp = Float64(x_m - Float64(Float64(x_m * z) / y));
	else
		tmp = fma(Float64(z / Float64(-y)), x_m, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -2e+79], N[(x$95$m - N[(N[(x$95$m * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / (-y)), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -1.99999999999999993e79

   1. Initial program 91.0%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.99999999999999993e79 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. associate-*r* N/A

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

      13. div-inv N/A

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

      14. *-inverses N/A

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

      15. *-commutative N/A

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

      16. *-rgt-identity N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lift-neg.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. distribute-neg-frac N/A

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

      24. lift-/.f64 N/A

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

      25. +-commutative N/A

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

   6. Applied rewrites 96.0%

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

Alternative 5: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y - z\right)}{y} \leq -2 \cdot 10^{+79}:\\ \;\;\;\;x\_m - \frac{x\_m \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y - z}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (- y z)) y) -2e+79)
    (- x_m (/ (* x_m z) y))
    (* x_m (/ (- y z) y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (y - z)) / y) <= -2e+79) {
		tmp = x_m - ((x_m * z) / y);
	} else {
		tmp = x_m * ((y - z) / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (y - z)) / y) <= (-2d+79)) then
        tmp = x_m - ((x_m * z) / y)
    else
        tmp = x_m * ((y - z) / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (y - z)) / y) <= -2e+79) {
		tmp = x_m - ((x_m * z) / y);
	} else {
		tmp = x_m * ((y - z) / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (y - z)) / y) <= -2e+79:
		tmp = x_m - ((x_m * z) / y)
	else:
		tmp = x_m * ((y - z) / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(y - z)) / y) <= -2e+79)
		tmp = Float64(x_m - Float64(Float64(x_m * z) / y));
	else
		tmp = Float64(x_m * Float64(Float64(y - z) / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((x_m * (y - z)) / y) <= -2e+79)
		tmp = x_m - ((x_m * z) / y);
	else
		tmp = x_m * ((y - z) / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -2e+79], N[(x$95$m - N[(N[(x$95$m * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -1.99999999999999993e79

   1. Initial program 91.0%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.99999999999999993e79 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -2 \cdot 10^{+79}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \frac{x\_m}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-29}:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (- z) (/ x_m y))))
   (* x_s (if (<= z -2.85e-36) t_0 (if (<= z 3.4e-29) (* x_m 1.0) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = -z * (x_m / y);
	double tmp;
	if (z <= -2.85e-36) {
		tmp = t_0;
	} else if (z <= 3.4e-29) {
		tmp = x_m * 1.0;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -z * (x_m / y)
    if (z <= (-2.85d-36)) then
        tmp = t_0
    else if (z <= 3.4d-29) then
        tmp = x_m * 1.0d0
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = -z * (x_m / y);
	double tmp;
	if (z <= -2.85e-36) {
		tmp = t_0;
	} else if (z <= 3.4e-29) {
		tmp = x_m * 1.0;
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = -z * (x_m / y)
	tmp = 0
	if z <= -2.85e-36:
		tmp = t_0
	elif z <= 3.4e-29:
		tmp = x_m * 1.0
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(-z) * Float64(x_m / y))
	tmp = 0.0
	if (z <= -2.85e-36)
		tmp = t_0;
	elseif (z <= 3.4e-29)
		tmp = Float64(x_m * 1.0);
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = -z * (x_m / y);
	tmp = 0.0;
	if (z <= -2.85e-36)
		tmp = t_0;
	elseif (z <= 3.4e-29)
		tmp = x_m * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[((-z) * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -2.85e-36], t$95$0, If[LessEqual[z, 3.4e-29], N[(x$95$m * 1.0), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8499999999999999e-36 or 3.39999999999999972e-29 < z

   1. Initial program 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      2. lower-neg.f64 74.4

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

   7. Applied rewrites 74.4%

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

   if -2.8499999999999999e-36 < z < 3.39999999999999972e-29

   1. Initial program 84.1%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      2. lower-neg.f64 19.2

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

   5. Applied rewrites 19.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{neg}\left(z\right)\right)}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(z\right)\right)}}{y} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\mathsf{neg}\left(z\right)}{y}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

      6. lower-/.f64 23.5

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

   7. Applied rewrites 23.5%

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

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 78.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-36}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-29}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m - \frac{x\_m \cdot z}{y}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (- x_m (/ (* x_m z) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m - ((x_m * z) / y));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m - ((x_m * z) / y))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m - ((x_m * z) / y));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (x_m - ((x_m * z) / y))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m - Float64(Float64(x_m * z) / y)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m - ((x_m * z) / y));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m - N[(N[(x$95$m * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

   \[\frac{x \cdot \left(y - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. *-inverses N/A

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

   4. distribute-lft-out-- N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 95.4

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

5. Applied rewrites 95.4%

   \[\leadsto \color{blue}{x - \frac{x \cdot z}{y}} \]
6. Add Preprocessing

Alternative 8: 50.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot 1\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* x_m 1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m * 1.0);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m * 1.0d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m * 1.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (x_m * 1.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m * 1.0))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m * 1.0);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

   \[\frac{x \cdot \left(y - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

      \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}{y} \]

   2. lower-neg.f64 50.7

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

5. Applied rewrites 50.7%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{neg}\left(z\right)\right)}{y}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(z\right)\right)}}{y} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{neg}\left(z\right)}{y}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z\right)}{y} \cdot x} \]

   6. lower-/.f64 48.7

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

7. Applied rewrites 48.7%

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

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{1} \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 47.6%

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

11. Final simplification 47.6%

    \[\leadsto x \cdot 1 \]
12. Add Preprocessing

Developer Target 1: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z -2.060202331921739e+104)
   (- x (/ (* z x) y))
   (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < -2.060202331921739e+104) {
		tmp = x - ((z * x) / y);
	} else if (z < 1.6939766013828526e+213) {
		tmp = x / (y / (y - z));
	} else {
		tmp = (y - z) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < (-2.060202331921739d+104)) then
        tmp = x - ((z * x) / y)
    else if (z < 1.6939766013828526d+213) then
        tmp = x / (y / (y - z))
    else
        tmp = (y - z) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < -2.060202331921739e+104) {
		tmp = x - ((z * x) / y);
	} else if (z < 1.6939766013828526e+213) {
		tmp = x / (y / (y - z));
	} else {
		tmp = (y - z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < -2.060202331921739e+104:
		tmp = x - ((z * x) / y)
	elif z < 1.6939766013828526e+213:
		tmp = x / (y / (y - z))
	else:
		tmp = (y - z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < -2.060202331921739e+104)
		tmp = Float64(x - Float64(Float64(z * x) / y));
	elseif (z < 1.6939766013828526e+213)
		tmp = Float64(x / Float64(y / Float64(y - z)));
	else
		tmp = Float64(Float64(y - z) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < -2.060202331921739e+104)
		tmp = x - ((z * x) / y);
	elseif (z < 1.6939766013828526e+213)
		tmp = x / (y / (y - z));
	else
		tmp = (y - z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, -2.060202331921739e+104], N[(x - N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Less[z, 1.6939766013828526e+213], N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024221 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -206020233192173900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (/ (* z x) y)) (if (< z 1693976601382852600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))

  (/ (* x (- y z)) y))