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

Percentage Accurate: 88.2% → 99.7%

Time: 6.5s

Alternatives: 9

Speedup: 0.8×

• 21.3% 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: 88.2% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(y - z\right)}{z} \cdot x\_m\\ \end{array} \end{array} \]

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 5e-34)
    (/ (fma (- y z) x_m x_m) z)
    (* (/ (+ 1.0 (- y z)) z) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e-34) {
		tmp = fma((y - z), x_m, x_m) / z;
	} else {
		tmp = ((1.0 + (y - z)) / z) * x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5e-34)
		tmp = Float64(fma(Float64(y - z), x_m, x_m) / z);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(y - z)) / 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, 5e-34], N[(N[(N[(y - z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000003e-34

   1. Initial program 91.4%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 91.4

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

   4. Applied rewrites 91.4%

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

   if 5.0000000000000003e-34 < x

   1. Initial program 77.0%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+196}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 (<= z -1.9e+196)
    (- x_m)
    (if (<= z 7.4e+171) (/ (fma (- y z) x_m 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 (z <= -1.9e+196) {
		tmp = -x_m;
	} else if (z <= 7.4e+171) {
		tmp = fma((y - z), x_m, 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 (z <= -1.9e+196)
		tmp = Float64(-x_m);
	elseif (z <= 7.4e+171)
		tmp = Float64(fma(Float64(y - z), x_m, x_m) / z);
	else
		tmp = Float64(Float64(x_m / 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[z, -1.9e+196], (-x$95$m), If[LessEqual[z, 7.4e+171], N[(N[(N[(y - z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.9000000000000001e196

   1. Initial program 47.2%

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

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

   5. Applied rewrites 92.4%

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

   if -1.9000000000000001e196 < z < 7.39999999999999996e171

   1. Initial program 96.2%

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

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

   4. Applied rewrites 96.2%

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

   if 7.39999999999999996e171 < z

   1. Initial program 61.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. *-inverses N/A

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 94.1

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

   5. Applied rewrites 94.1%

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

Alternative 3: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{z} \cdot x\_m\\ \mathbf{elif}\;y \leq 380000:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x\_m, x\_m\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -2.2e+15)
    (* (/ y z) x_m)
    (if (<= y 380000.0) (- (/ x_m z) x_m) (/ (fma y 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 (y <= -2.2e+15) {
		tmp = (y / z) * x_m;
	} else if (y <= 380000.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = fma(y, x_m, x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -2.2e+15)
		tmp = Float64(Float64(y / z) * x_m);
	elseif (y <= 380000.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(fma(y, x_m, x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -2.2e+15], N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[y, 380000.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(y * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e15

   1. Initial program 87.7%

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

      1. lift-/.f64 N/A

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

      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 95.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

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

   if -2.2e15 < y < 3.8e5

   1. Initial program 84.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. *-inverses N/A

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 3.8e5 < y

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 80.2

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

   5. Applied rewrites 80.2%

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

Alternative 4: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+15} \lor \neg \left(y \leq 380000\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 -2.2e+15) (not (<= y 380000.0)))
    (* (/ 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 <= -2.2e+15) || !(y <= 380000.0)) {
		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 <= (-2.2d+15)) .or. (.not. (y <= 380000.0d0))) 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 <= -2.2e+15) || !(y <= 380000.0)) {
		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 <= -2.2e+15) or not (y <= 380000.0):
		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 <= -2.2e+15) || !(y <= 380000.0))
		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 <= -2.2e+15) || ~((y <= 380000.0)))
		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, -2.2e+15], N[Not[LessEqual[y, 380000.0]], $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 < -2.2e15 or 3.8e5 < y

   1. Initial program 89.7%

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

      1. lift-/.f64 N/A

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

      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 93.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 93.0

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   6. 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 76.9

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

   7. Applied rewrites 76.9%

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

   if -2.2e15 < y < 3.8e5

   1. Initial program 84.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. *-inverses N/A

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+15} \lor \neg \left(y \leq 380000\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{z} \cdot x\_m\\ \mathbf{elif}\;y \leq 380000:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -2.2e+15)
    (* (/ y z) x_m)
    (if (<= y 380000.0) (- (/ x_m z) x_m) (/ (* y x_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.2e+15) {
		tmp = (y / z) * x_m;
	} else if (y <= 380000.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (y * x_m) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.2d+15)) then
        tmp = (y / z) * x_m
    else if (y <= 380000.0d0) then
        tmp = (x_m / z) - x_m
    else
        tmp = (y * x_m) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.2e+15) {
		tmp = (y / z) * x_m;
	} else if (y <= 380000.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (y * x_m) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -2.2e+15:
		tmp = (y / z) * x_m
	elif y <= 380000.0:
		tmp = (x_m / z) - x_m
	else:
		tmp = (y * x_m) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -2.2e+15)
		tmp = Float64(Float64(y / z) * x_m);
	elseif (y <= 380000.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(Float64(y * x_m) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -2.2e+15)
		tmp = (y / z) * x_m;
	elseif (y <= 380000.0)
		tmp = (x_m / z) - x_m;
	else
		tmp = (y * x_m) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -2.2e+15], N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[y, 380000.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e15

   1. Initial program 87.7%

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

      1. lift-/.f64 N/A

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

      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 95.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

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

   if -2.2e15 < y < 3.8e5

   1. Initial program 84.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. *-inverses N/A

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 3.8e5 < y

   1. Initial program 91.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

Alternative 6: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{z} \cdot x\_m\\ \mathbf{elif}\;y \leq 380000:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -2.2e+15)
    (* (/ y z) x_m)
    (if (<= y 380000.0) (- (/ x_m z) x_m) (* (/ x_m z) y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.2e+15) {
		tmp = (y / z) * x_m;
	} else if (y <= 380000.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (x_m / z) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.2d+15)) then
        tmp = (y / z) * x_m
    else if (y <= 380000.0d0) then
        tmp = (x_m / z) - x_m
    else
        tmp = (x_m / z) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -2.2e+15) {
		tmp = (y / z) * x_m;
	} else if (y <= 380000.0) {
		tmp = (x_m / z) - x_m;
	} else {
		tmp = (x_m / z) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -2.2e+15:
		tmp = (y / z) * x_m
	elif y <= 380000.0:
		tmp = (x_m / z) - x_m
	else:
		tmp = (x_m / z) * y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -2.2e+15)
		tmp = Float64(Float64(y / z) * x_m);
	elseif (y <= 380000.0)
		tmp = Float64(Float64(x_m / z) - x_m);
	else
		tmp = Float64(Float64(x_m / z) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -2.2e+15)
		tmp = (y / z) * x_m;
	elseif (y <= 380000.0)
		tmp = (x_m / z) - x_m;
	else
		tmp = (x_m / z) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -2.2e+15], N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[y, 380000.0], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e15

   1. Initial program 87.7%

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

      1. lift-/.f64 N/A

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

      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 95.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

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

   if -2.2e15 < y < 3.8e5

   1. Initial program 84.1%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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. *-inverses N/A

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

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

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 3.8e5 < y

   1. Initial program 91.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 90.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 90.1

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   6. 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 77.4

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

   7. Applied rewrites 77.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;y \leq 380000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.95 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, x\_m, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(1 + \left(y - z\right)\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.95e-30)
    (/ (fma (- y z) x_m x_m) z)
    (* (/ x_m z) (+ 1.0 (- y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.95e-30) {
		tmp = fma((y - z), x_m, x_m) / z;
	} else {
		tmp = (x_m / z) * (1.0 + (y - z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.95e-30)
		tmp = Float64(fma(Float64(y - z), x_m, x_m) / z);
	else
		tmp = Float64(Float64(x_m / z) * Float64(1.0 + Float64(y - z)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.95e-30], N[(N[(N[(y - z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.9500000000000002e-30

   1. Initial program 91.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   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 91.5

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

   4. Applied rewrites 91.5%

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

   if 1.9500000000000002e-30 < x

   1. Initial program 76.4%

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

      1. lift-/.f64 N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 8: 65.0% 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 87.1%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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. *-inverses N/A

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

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

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

   5. *-rgt-identity N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 60.0

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

5. Applied rewrites 60.0%

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

Alternative 9: 38.8% 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 87.1%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
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 34.4

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

5. Applied rewrites 34.4%

   \[\leadsto \color{blue}{-x} \]
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 2024339 
(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))