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

Percentage Accurate: 88.0% → 99.8%

Time: 7.1s

Alternatives: 8

Speedup: 1.1×

• 21.4% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{\left(y - z\right) - -1}}\\ \end{array} \end{array} \]

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 2e+35)
    (- (/ (fma y x_m 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 <= 2e+35) {
		tmp = (fma(y, x_m, 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 <= 2e+35)
		tmp = Float64(Float64(fma(y, x_m, 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, 2e+35], N[(N[(N[(y * x$95$m + 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 < 1.9999999999999999e35

   1. Initial program 91.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. associate-/l* N/A

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

      9. div-sub N/A

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

      10. *-inverses N/A

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

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

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

      12. associate-/l* N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

   10. Applied rewrites 94.9%

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

   if 1.9999999999999999e35 < x

   1. Initial program 76.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+68} \lor \neg \left(y \leq 4.8 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 -5e+68) (not (<= y 4.8e+103)))
    (* (/ 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 <= -5e+68) || !(y <= 4.8e+103)) {
		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 <= (-5d+68)) .or. (.not. (y <= 4.8d+103))) 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 <= -5e+68) || !(y <= 4.8e+103)) {
		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 <= -5e+68) or not (y <= 4.8e+103):
		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 <= -5e+68) || !(y <= 4.8e+103))
		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 <= -5e+68) || ~((y <= 4.8e+103)))
		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, -5e+68], N[Not[LessEqual[y, 4.8e+103]], $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 < -5.0000000000000004e68 or 4.7999999999999997e103 < y

   1. Initial program 87.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -5.0000000000000004e68 < y < 4.7999999999999997e103

   1. Initial program 87.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+68} \lor \neg \left(y \leq 4.8 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z\right) - -1}{z} \cdot x\_m\\ \end{array} \end{array} \]

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 (<= x_m 2e+38)
    (- (/ (fma y x_m x_m) z) x_m)
    (* (/ (- (- y z) -1.0) 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 <= 2e+38) {
		tmp = (fma(y, x_m, x_m) / z) - x_m;
	} else {
		tmp = (((y - z) - -1.0) / 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 <= 2e+38)
		tmp = Float64(Float64(fma(y, x_m, x_m) / z) - x_m);
	else
		tmp = Float64(Float64(Float64(Float64(y - z) - -1.0) / z) * 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, 2e+38], N[(N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(N[(N[(y - z), $MachinePrecision] - -1.0), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999995e38

   1. Initial program 91.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.9

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

   7. Applied rewrites 54.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. associate-/l* N/A

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

      9. div-sub N/A

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

      10. *-inverses N/A

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

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

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

      12. associate-/l* N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

   10. Applied rewrites 95.0%

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

   if 1.99999999999999995e38 < x

   1. Initial program 76.0%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. associate--r+ N/A

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

      13. lift--.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - z\right) - -1}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(\left(y - z\right) - -1\right)\\ \end{array} \end{array} \]

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 5.2e-59)
    (/ (fma (- y z) x_m x_m) z)
    (* (/ 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 <= 5.2e-59) {
		tmp = fma((y - z), x_m, x_m) / z;
	} 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 <= 5.2e-59)
		tmp = Float64(fma(Float64(y - z), x_m, x_m) / z);
	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, 5.2e-59], N[(N[(N[(y - z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] / z), $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 < 5.19999999999999996e-59

   1. Initial program 90.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 90.7

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

   4. Applied rewrites 90.7%

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

   if 5.19999999999999996e-59 < 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 99.9

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

      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 99.9

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

   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. Add Preprocessing

Alternative 5: 65.5% accurate, 1.0× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 75.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 75.1

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

   5. Applied rewrites 75.1%

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

   if -1 < z < 1

   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

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - z\right)}{z}} \]
   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 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 50.7%

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

4. Final simplification 63.3%

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

Alternative 6: 95.9% accurate, 1.1× speedup

Math

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

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 y 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) {
	return x_s * ((fma(y, x_m, 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(y, x_m, 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[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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 87.4

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

4. Applied rewrites 87.4%

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

5. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 55.9

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

7. Applied rewrites 55.9%

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

8. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-lft-in N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

   8. associate-/l* N/A

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

   9. div-sub N/A

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

   10. *-inverses N/A

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

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

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

   12. associate-/l* N/A

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

   13. *-rgt-identity N/A

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

   14. distribute-lft-in N/A

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

   15. *-rgt-identity N/A

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

   16. lower--.f64 N/A

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

10. Applied rewrites 95.1%

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

11. Final simplification 95.1%

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

Alternative 7: 66.6% 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 87.4%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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 64.5

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

5. Applied rewrites 64.5%

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

Alternative 8: 39.7% 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 87.4%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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 40.3

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

5. Applied rewrites 40.3%

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

Developer Target 1: 99.4% 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 2024271 
(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))