Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.0% → 99.9%

Time: 10.5s

Alternatives: 12

Speedup: 1.1×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + 1.0d0)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + 1, \frac{x\_m}{z}, -x\_m\right)\\ \end{array} \end{array} \]

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 5e+16)
    (- (/ (fma x_m y x_m) z) x_m)
    (fma (+ y 1.0) (/ x_m z) (- 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 <= 5e+16) {
		tmp = (fma(x_m, y, x_m) / z) - x_m;
	} else {
		tmp = fma((y + 1.0), (x_m / z), -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 (x_m <= 5e+16)
		tmp = Float64(Float64(fma(x_m, y, x_m) / z) - x_m);
	else
		tmp = fma(Float64(y + 1.0), Float64(x_m / z), Float64(-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[x$95$m, 5e+16], N[(N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(y + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision] + (-x$95$m)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e16

   1. Initial program 94.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 98.0%

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

   if 5e16 < x

   1. Initial program 74.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 91.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 2: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot y}{z} - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (- (/ (* x_m y) z) x_m)))
   (* x_s (if (<= y -1.0) t_0 (if (<= y 7.8e-7) (- (/ x_m z) x_m) 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 = ((x_m * y) / z) - x_m;
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 7.8e-7) {
		tmp = (x_m / z) - x_m;
	} 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 = ((x_m * y) / z) - x_m
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 7.8d-7) then
        tmp = (x_m / z) - x_m
    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 = ((x_m * y) / z) - x_m;
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 7.8e-7) {
		tmp = (x_m / z) - x_m;
	} 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 = ((x_m * y) / z) - x_m
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 7.8e-7:
		tmp = (x_m / z) - x_m
	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(Float64(x_m * y) / z) - x_m)
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 7.8e-7)
		tmp = Float64(Float64(x_m / z) - x_m);
	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 = ((x_m * y) / z) - x_m;
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 7.8e-7)
		tmp = (x_m / z) - x_m;
	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[(N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 7.8e-7], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 7.80000000000000049e-7 < y

   1. Initial program 90.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 92.9

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

   8. Applied rewrites 92.9%

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

   if -1 < y < 7.80000000000000049e-7

   1. Initial program 89.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 3: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -0.49:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z}\\ \mathbf{elif}\;y \leq 190000000000:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \end{array} \end{array} \]

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 (<= y -0.49)
    (/ (fma x_m y x_m) z)
    (if (<= y 190000000000.0) (- (/ x_m z) x_m) (/ (* x_m 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 (y <= -0.49) {
		tmp = fma(x_m, y, x_m) / z;
	} else if (y <= 190000000000.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (x_m * 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 (y <= -0.49)
		tmp = Float64(fma(x_m, y, x_m) / z);
	elseif (y <= 190000000000.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(Float64(x_m * y) / z);
	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[y, -0.49], N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 190000000000.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -0.48999999999999999

   1. Initial program 93.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -0.48999999999999999 < y < 1.9e11

   1. Initial program 89.3%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 1.9e11 < y

   1. Initial program 87.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

Alternative 4: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -26000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 190000000000:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (/ (* x_m y) z)))
   (*
    x_s
    (if (<= y -26000.0)
      t_0
      (if (<= y 190000000000.0) (- (/ x_m z) x_m) 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 = (x_m * y) / z;
	double tmp;
	if (y <= -26000.0) {
		tmp = t_0;
	} else if (y <= 190000000000.0) {
		tmp = (x_m / z) - x_m;
	} 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 = (x_m * y) / z
    if (y <= (-26000.0d0)) then
        tmp = t_0
    else if (y <= 190000000000.0d0) then
        tmp = (x_m / z) - x_m
    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 = (x_m * y) / z;
	double tmp;
	if (y <= -26000.0) {
		tmp = t_0;
	} else if (y <= 190000000000.0) {
		tmp = (x_m / z) - x_m;
	} 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 = (x_m * y) / z
	tmp = 0
	if y <= -26000.0:
		tmp = t_0
	elif y <= 190000000000.0:
		tmp = (x_m / z) - x_m
	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(x_m * y) / z)
	tmp = 0.0
	if (y <= -26000.0)
		tmp = t_0;
	elseif (y <= 190000000000.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	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 = (x_m * y) / z;
	tmp = 0.0;
	if (y <= -26000.0)
		tmp = t_0;
	elseif (y <= 190000000000.0)
		tmp = (x_m / z) - x_m;
	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[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -26000.0], t$95$0, If[LessEqual[y, 190000000000.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -26000 or 1.9e11 < y

   1. Initial program 90.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if -26000 < y < 1.9e11

   1. Initial program 89.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 5: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -26000:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 190000000000:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

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 (<= y -26000.0)
    (* x_m (/ y z))
    (if (<= y 190000000000.0) (- (/ x_m z) x_m) (* y (/ x_m 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 (y <= -26000.0) {
		tmp = x_m * (y / z);
	} else if (y <= 190000000000.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y * (x_m / 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 (y <= (-26000.0d0)) then
        tmp = x_m * (y / z)
    else if (y <= 190000000000.0d0) then
        tmp = (x_m / z) - x_m
    else
        tmp = y * (x_m / 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 (y <= -26000.0) {
		tmp = x_m * (y / z);
	} else if (y <= 190000000000.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = y * (x_m / 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 y <= -26000.0:
		tmp = x_m * (y / z)
	elif y <= 190000000000.0:
		tmp = (x_m / z) - x_m
	else:
		tmp = y * (x_m / 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 (y <= -26000.0)
		tmp = Float64(x_m * Float64(y / z));
	elseif (y <= 190000000000.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(y * Float64(x_m / 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 (y <= -26000.0)
		tmp = x_m * (y / z);
	elseif (y <= 190000000000.0)
		tmp = (x_m / z) - x_m;
	else
		tmp = y * (x_m / 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[y, -26000.0], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 190000000000.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -26000

   1. Initial program 93.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 77.8

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

   7. Applied rewrites 77.8%

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

   if -26000 < y < 1.9e11

   1. Initial program 89.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 1.9e11 < y

   1. Initial program 87.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -26000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 190000000000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -26000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 190000000000:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* y (/ x_m z))))
   (*
    x_s
    (if (<= y -26000.0)
      t_0
      (if (<= y 190000000000.0) (- (/ x_m z) x_m) 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 = y * (x_m / z);
	double tmp;
	if (y <= -26000.0) {
		tmp = t_0;
	} else if (y <= 190000000000.0) {
		tmp = (x_m / z) - x_m;
	} 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 = y * (x_m / z)
    if (y <= (-26000.0d0)) then
        tmp = t_0
    else if (y <= 190000000000.0d0) then
        tmp = (x_m / z) - x_m
    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 = y * (x_m / z);
	double tmp;
	if (y <= -26000.0) {
		tmp = t_0;
	} else if (y <= 190000000000.0) {
		tmp = (x_m / z) - x_m;
	} 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 = y * (x_m / z)
	tmp = 0
	if y <= -26000.0:
		tmp = t_0
	elif y <= 190000000000.0:
		tmp = (x_m / z) - x_m
	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(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -26000.0)
		tmp = t_0;
	elseif (y <= 190000000000.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	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 = y * (x_m / z);
	tmp = 0.0;
	if (y <= -26000.0)
		tmp = t_0;
	elseif (y <= 190000000000.0)
		tmp = (x_m / z) - x_m;
	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[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -26000.0], t$95$0, If[LessEqual[y, 190000000000.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -26000 or 1.9e11 < y

   1. Initial program 90.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 75.4

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

   7. Applied rewrites 75.4%

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

   if -26000 < y < 1.9e11

   1. Initial program 89.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 7: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{1 + \left(y - z\right)}{z}\\ \end{array} \end{array} \]

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 1e+25)
    (- (/ (fma x_m y x_m) z) x_m)
    (* x_m (/ (+ 1.0 (- y z)) 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 <= 1e+25) {
		tmp = (fma(x_m, y, x_m) / z) - x_m;
	} else {
		tmp = x_m * ((1.0 + (y - z)) / 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 (x_m <= 1e+25)
		tmp = Float64(Float64(fma(x_m, y, x_m) / z) - x_m);
	else
		tmp = Float64(x_m * Float64(Float64(1.0 + Float64(y - z)) / z));
	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[x$95$m, 1e+25], N[(N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(x$95$m * N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000009e25

   1. Initial program 94.3%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 98.1%

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

   if 1.00000000000000009e25 < x

   1. Initial program 73.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 + \left(y - z\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(1 + \left(y - z\right)\right)\\ \end{array} \end{array} \]

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 5e+16)
    (- (/ (fma x_m y x_m) z) x_m)
    (* (/ x_m z) (+ 1.0 (- 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 <= 5e+16) {
		tmp = (fma(x_m, y, x_m) / z) - x_m;
	} else {
		tmp = (x_m / z) * (1.0 + (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 (x_m <= 5e+16)
		tmp = Float64(Float64(fma(x_m, y, x_m) / z) - x_m);
	else
		tmp = Float64(Float64(x_m / z) * Float64(1.0 + Float64(y - z)));
	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[x$95$m, 5e+16], N[(N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e16

   1. Initial program 94.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 98.0%

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

   if 5e16 < x

   1. Initial program 74.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

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 (<= z -13.0) (- x_m) (if (<= z 6e-7) (/ x_m z) (- 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 (z <= -13.0) {
		tmp = -x_m;
	} else if (z <= 6e-7) {
		tmp = x_m / z;
	} else {
		tmp = -x_m;
	}
	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 (z <= (-13.0d0)) then
        tmp = -x_m
    else if (z <= 6d-7) then
        tmp = x_m / z
    else
        tmp = -x_m
    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 (z <= -13.0) {
		tmp = -x_m;
	} else if (z <= 6e-7) {
		tmp = x_m / z;
	} else {
		tmp = -x_m;
	}
	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 z <= -13.0:
		tmp = -x_m
	elif z <= 6e-7:
		tmp = x_m / z
	else:
		tmp = -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 (z <= -13.0)
		tmp = Float64(-x_m);
	elseif (z <= 6e-7)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(-x_m);
	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 (z <= -13.0)
		tmp = -x_m;
	elseif (z <= 6e-7)
		tmp = x_m / z;
	else
		tmp = -x_m;
	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[z, -13.0], (-x$95$m), If[LessEqual[z, 6e-7], N[(x$95$m / z), $MachinePrecision], (-x$95$m)]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -13 or 5.9999999999999997e-7 < z

   1. Initial program 80.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if -13 < z < 5.9999999999999997e-7

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 54.1

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

   8. Applied rewrites 54.1%

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

Alternative 10: 95.7% accurate, 1.1× speedup

Math

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

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 (- (/ (fma x_m y x_m) z) 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) {
	return x_s * ((fma(x_m, y, x_m) / z) - x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(fma(x_m, y, x_m) / z) - x_m))
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[(N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. *-inverses N/A

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

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

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

   5. associate-/l* N/A

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

   6. *-rgt-identity N/A

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

   7. lower--.f64 N/A

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

5. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]
6. Add Preprocessing

Alternative 11: 65.8% accurate, 1.5× speedup

Math

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

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 z) 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) {
	return x_s * ((x_m / z) - x_m);
}

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 / z) - x_m)
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 / z) - x_m);
}

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 / z) - x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(x_m / z) - x_m))
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 / z) - x_m);
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[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. *-inverses N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-in N/A

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

   7. associate-/l* N/A

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

   8. *-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-/.f64 63.0

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

5. Applied rewrites 63.0%

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

Alternative 12: 38.6% accurate, 7.7× speedup

Math

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

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)))

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;
}

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
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;
}

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

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))
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;
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 * (-x$95$m)), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 37.3

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

5. Applied rewrites 37.3%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -67870776678359/25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (+ 1 y) (/ x z)) x) (if (< x 1937054408219773/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x))))

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