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

Percentage Accurate: 88.6% → 99.8%

Time: 7.2s

Alternatives: 8

Speedup: 1.1×

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: 88.6% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{\left(y - z\right) - -1}}\\ \end{array} \end{array} \]

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 4.5e-62)
    (- (/ (fma x_m y x_m) z) x_m)
    (/ x_m (/ z (- (- y z) -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) {
	double tmp;
	if (x_m <= 4.5e-62) {
		tmp = (fma(x_m, y, x_m) / z) - x_m;
	} else {
		tmp = x_m / (z / ((y - z) - -1.0));
	}
	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 <= 4.5e-62)
		tmp = Float64(Float64(fma(x_m, y, x_m) / z) - x_m);
	else
		tmp = Float64(x_m / Float64(z / Float64(Float64(y - z) - -1.0)));
	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, 4.5e-62], N[(N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(x$95$m / N[(z / N[(N[(y - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.50000000000000018e-62

   1. Initial program 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 92.5

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

   4. Applied rewrites 92.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-out N/A

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

      16. lift-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. unsub-neg N/A

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

      20. lift-/.f64 N/A

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

      21. div-inv N/A

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

      22. associate-*l* N/A

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

      23. inv-pow N/A

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

      24. pow-plus N/A

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

   6. Applied rewrites 98.8%

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

   if 4.50000000000000018e-62 < x

   1. Initial program 83.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+l- N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. associate--r+ N/A

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

      14. lift--.f64 N/A

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

      15. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 2: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.032\right):\\ \;\;\;\;\frac{y \cdot x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 (or (<= y -1.0) (not (<= y 0.032)))
    (- (/ (* y x_m) z) x_m)
    (- (/ 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 ((y <= -1.0) || !(y <= 0.032)) {
		tmp = ((y * x_m) / z) - x_m;
	} else {
		tmp = (x_m / z) - 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 ((y <= (-1.0d0)) .or. (.not. (y <= 0.032d0))) then
        tmp = ((y * x_m) / z) - x_m
    else
        tmp = (x_m / z) - 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 ((y <= -1.0) || !(y <= 0.032)) {
		tmp = ((y * x_m) / z) - x_m;
	} else {
		tmp = (x_m / z) - 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 (y <= -1.0) or not (y <= 0.032):
		tmp = ((y * x_m) / z) - x_m
	else:
		tmp = (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 ((y <= -1.0) || !(y <= 0.032))
		tmp = Float64(Float64(Float64(y * x_m) / z) - x_m);
	else
		tmp = Float64(Float64(x_m / z) - 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 ((y <= -1.0) || ~((y <= 0.032)))
		tmp = ((y * x_m) / z) - x_m;
	else
		tmp = (x_m / z) - 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[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.032]], $MachinePrecision]], N[(N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.032000000000000001 < y

   1. Initial program 90.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-out N/A

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

      16. lift-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. unsub-neg N/A

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

      20. lift-/.f64 N/A

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

      21. div-inv N/A

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

      22. associate-*l* N/A

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

      23. inv-pow N/A

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

      24. pow-plus N/A

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 95.5

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

   9. Applied rewrites 95.5%

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

   if -1 < y < 0.032000000000000001

   1. Initial program 89.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.032\right):\\ \;\;\;\;\frac{y \cdot x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+35} \lor \neg \left(y \leq 4.7 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{y \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 (or (<= y -1.05e+35) (not (<= y 4.7e+98)))
    (/ (* y x_m) z)
    (- (/ 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 ((y <= -1.05e+35) || !(y <= 4.7e+98)) {
		tmp = (y * x_m) / z;
	} else {
		tmp = (x_m / z) - 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 ((y <= (-1.05d+35)) .or. (.not. (y <= 4.7d+98))) then
        tmp = (y * x_m) / z
    else
        tmp = (x_m / z) - 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 ((y <= -1.05e+35) || !(y <= 4.7e+98)) {
		tmp = (y * x_m) / z;
	} else {
		tmp = (x_m / z) - 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 (y <= -1.05e+35) or not (y <= 4.7e+98):
		tmp = (y * x_m) / z
	else:
		tmp = (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 ((y <= -1.05e+35) || !(y <= 4.7e+98))
		tmp = Float64(Float64(y * x_m) / z);
	else
		tmp = Float64(Float64(x_m / z) - 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 ((y <= -1.05e+35) || ~((y <= 4.7e+98)))
		tmp = (y * x_m) / z;
	else
		tmp = (x_m / z) - 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[Or[LessEqual[y, -1.05e+35], N[Not[LessEqual[y, 4.7e+98]], $MachinePrecision]], N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.0499999999999999e35 or 4.6999999999999997e98 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if -1.0499999999999999e35 < y < 4.6999999999999997e98

   1. Initial program 88.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+35} \lor \neg \left(y \leq 4.7 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+35} \lor \neg \left(y \leq 3.3 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 (or (<= y -1.05e+35) (not (<= y 3.3e+98)))
    (* (/ x_m z) 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) {
	double tmp;
	if ((y <= -1.05e+35) || !(y <= 3.3e+98)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = (x_m / z) - 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 ((y <= (-1.05d+35)) .or. (.not. (y <= 3.3d+98))) then
        tmp = (x_m / z) * y
    else
        tmp = (x_m / z) - 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 ((y <= -1.05e+35) || !(y <= 3.3e+98)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = (x_m / z) - 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 (y <= -1.05e+35) or not (y <= 3.3e+98):
		tmp = (x_m / z) * y
	else:
		tmp = (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 ((y <= -1.05e+35) || !(y <= 3.3e+98))
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(Float64(x_m / z) - 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 ((y <= -1.05e+35) || ~((y <= 3.3e+98)))
		tmp = (x_m / z) * y;
	else
		tmp = (x_m / z) - 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[Or[LessEqual[y, -1.05e+35], N[Not[LessEqual[y, 3.3e+98]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.0499999999999999e35 or 3.30000000000000028e98 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   4. 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 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 75.0%

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

   if -1.0499999999999999e35 < y < 3.30000000000000028e98

   1. Initial program 88.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+35} \lor \neg \left(y \leq 3.3 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(\left(y - z\right) - -1\right)\\ \end{array} \end{array} \]

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 1.5e-18)
    (- (/ (fma x_m y x_m) z) x_m)
    (* (/ x_m z) (- (- y z) -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) {
	double tmp;
	if (x_m <= 1.5e-18) {
		tmp = (fma(x_m, y, x_m) / z) - x_m;
	} else {
		tmp = (x_m / z) * ((y - z) - -1.0);
	}
	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 <= 1.5e-18)
		tmp = Float64(Float64(fma(x_m, y, x_m) / z) - x_m);
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y - z) - -1.0));
	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, 1.5e-18], 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[(N[(y - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.49999999999999991e-18

   1. Initial program 93.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 93.0

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

   4. Applied rewrites 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-out N/A

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

      16. lift-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. unsub-neg N/A

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

      20. lift-/.f64 N/A

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

      21. div-inv N/A

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

      22. associate-*l* N/A

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

      23. inv-pow N/A

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

      24. pow-plus N/A

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

   6. Applied rewrites 98.9%

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

   if 1.49999999999999991e-18 < x

   1. Initial program 80.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-+l- N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. associate--r+ N/A

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

      14. lift--.f64 N/A

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

      15. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 6: 95.9% 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 89.9%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 89.9

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

4. Applied rewrites 89.9%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-lft1-in N/A

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

   4. lift--.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l/ N/A

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

   7. lift-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. sub-neg N/A

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

   12. lift-neg.f64 N/A

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

   13. associate-+r+ N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-lft-out N/A

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

   16. lift-*.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. distribute-rgt-neg-out N/A

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

   19. unsub-neg N/A

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

   20. lift-/.f64 N/A

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

   21. div-inv N/A

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

   22. associate-*l* N/A

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

   23. inv-pow N/A

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

   24. pow-plus N/A

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

6. Applied rewrites 98.1%

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

Alternative 7: 65.9% 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 89.9%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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 70.3

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

5. Applied rewrites 70.3%

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

Alternative 8: 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 89.9%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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 45.3

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

5. Applied rewrites 45.3%

   \[\leadsto \color{blue}{-x} \]
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 2024313 
(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))