Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.3% → 96.6%

Time: 7.3s

Alternatives: 5

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * (y + z)) / 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)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * (y + z)) / 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)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(z + y\right) \cdot x\_m}{z} \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z} \cdot y, x\_m, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (+ z y) x_m) z) -2e+27)
    (/ (* y x_m) z)
    (fma (* (/ 1.0 z) y) x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((z + y) * x_m) / z) <= -2e+27) {
		tmp = (y * x_m) / z;
	} else {
		tmp = fma(((1.0 / z) * y), x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(z + y) * x_m) / z) <= -2e+27)
		tmp = Float64(Float64(y * x_m) / z);
	else
		tmp = fma(Float64(Float64(1.0 / z) * y), x_m, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[(z + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], -2e+27], N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * y), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -2e27

   1. Initial program 97.9%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   if -2e27 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 83.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. div-sub N/A

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

      7. *-inverses N/A

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

      8. associate-*r/ N/A

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

      9. distribute-rgt-out-- N/A

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

      10. *-lft-identity N/A

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

      11. mul-1-neg N/A

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

      12. cancel-sign-sub N/A

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

      13. distribute-rgt1-in N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z + y\right) \cdot x}{z} \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z} \cdot y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\left(z + y\right) \cdot x\_m}{z}\\ t_1 := \frac{y \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-234}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+220}:\\ \;\;\;\;-1 \cdot \left(-x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* (+ z y) x_m) z)) (t_1 (/ (* y x_m) z)))
   (*
    x_s
    (if (<= t_0 -2e-234) t_1 (if (<= t_0 1e+220) (* -1.0 (- x_m)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = ((z + y) * x_m) / z;
	double t_1 = (y * x_m) / z;
	double tmp;
	if (t_0 <= -2e-234) {
		tmp = t_1;
	} else if (t_0 <= 1e+220) {
		tmp = -1.0 * -x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((z + y) * x_m) / z
    t_1 = (y * x_m) / z
    if (t_0 <= (-2d-234)) then
        tmp = t_1
    else if (t_0 <= 1d+220) then
        tmp = (-1.0d0) * -x_m
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = ((z + y) * x_m) / z;
	double t_1 = (y * x_m) / z;
	double tmp;
	if (t_0 <= -2e-234) {
		tmp = t_1;
	} else if (t_0 <= 1e+220) {
		tmp = -1.0 * -x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = ((z + y) * x_m) / z
	t_1 = (y * x_m) / z
	tmp = 0
	if t_0 <= -2e-234:
		tmp = t_1
	elif t_0 <= 1e+220:
		tmp = -1.0 * -x_m
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(Float64(z + y) * x_m) / z)
	t_1 = Float64(Float64(y * x_m) / z)
	tmp = 0.0
	if (t_0 <= -2e-234)
		tmp = t_1;
	elseif (t_0 <= 1e+220)
		tmp = Float64(-1.0 * Float64(-x_m));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = ((z + y) * x_m) / z;
	t_1 = (y * x_m) / z;
	tmp = 0.0;
	if (t_0 <= -2e-234)
		tmp = t_1;
	elseif (t_0 <= 1e+220)
		tmp = -1.0 * -x_m;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z + y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-234], t$95$1, If[LessEqual[t$95$0, 1e+220], N[(-1.0 * (-x$95$m)), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -1.9999999999999999e-234 or 1e220 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 75.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 68.8

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

   5. Applied rewrites 68.8%

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

   if -1.9999999999999999e-234 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1e220

   1. Initial program 92.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 72.6%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z + y\right) \cdot x}{z} \leq -2 \cdot 10^{-234}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{\left(z + y\right) \cdot x}{z} \leq 10^{+220}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{z}{y + z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ x (/ z (+ y z))))

C

double code(double x, double y, double z) {
	return x / (z / (y + 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 / (z / (y + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ z (+ y z))))

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