Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Accuracy: 67.6% → 82.1%

Time: 8.5s

Precision: binary64

Cost: 1481

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-24} \lor \neg \left(t_0 \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)))
   (if (or (<= t_0 2e-24) (not (<= t_0 2e+297))) (+ x (/ x (/ z y))) t_0)))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if ((t_0 <= 2e-24) || !(t_0 <= 2e+297)) {
		tmp = x + (x / (z / y));
	} 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
    code = (x * (y + z)) / z
end function

↓

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 = (x * (y + z)) / z
    if ((t_0 <= 2d-24) .or. (.not. (t_0 <= 2d+297))) then
        tmp = x + (x / (z / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if ((t_0 <= 2e-24) || !(t_0 <= 2e+297)) {
		tmp = x + (x / (z / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * (y + z)) / z
	tmp = 0
	if (t_0 <= 2e-24) or not (t_0 <= 2e+297):
		tmp = x + (x / (z / y))
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(y + z)) / z)
	tmp = 0.0
	if ((t_0 <= 2e-24) || !(t_0 <= 2e+297))
		tmp = Float64(x + Float64(x / Float64(z / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * (y + z)) / z;
	tmp = 0.0;
	if ((t_0 <= 2e-24) || ~((t_0 <= 2e+297)))
		tmp = x + (x / (z / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-24], N[Not[LessEqual[t$95$0, 2e+297]], $MachinePrecision]], N[(x + N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Target

Original	67.6%
Target	80.2%
Herbie	82.1%

\[\frac{x}{\frac{z}{y + z}} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 1.99999999999999985e-24 or 2e297 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 59.9%

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

   2. Simplified 77.7%

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

      [Start] 59.9	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 63.6	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
      distribute-rgt-in [=>] 63.6	\[ \color{blue}{y \cdot \frac{x}{z} + z \cdot \frac{x}{z}} \]
      *-commutative [=>] 63.6	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{z} \cdot z} \]
      associate-/r/ [<=] 75.5	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{\frac{z}{z}}} \]
      *-inverses [=>] 75.5	\[ y \cdot \frac{x}{z} + \frac{x}{\color{blue}{1}} \]
      /-rgt-identity [=>] 75.5	\[ y \cdot \frac{x}{z} + \color{blue}{x} \]
      associate-*r/ [=>] 73.0	\[ \color{blue}{\frac{y \cdot x}{z}} + x \]
      *-commutative [<=] 73.0	\[ \frac{\color{blue}{x \cdot y}}{z} + x \]
      associate-*r/ [<=] 77.7	\[ \color{blue}{x \cdot \frac{y}{z}} + x \]
      fma-def [=>] 77.7	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]

   3. Applied egg-rr 77.8%

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

      [Start] 77.7	\[ \mathsf{fma}\left(x, \frac{y}{z}, x\right) \]
      fma-udef [=>] 77.7	\[ \color{blue}{x \cdot \frac{y}{z} + x} \]
      clear-num [=>] 77.6	\[ x \cdot \color{blue}{\frac{1}{\frac{z}{y}}} + x \]
      un-div-inv [=>] 77.8	\[ \color{blue}{\frac{x}{\frac{z}{y}}} + x \]

   if 1.99999999999999985e-24 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2e297

   1. Initial program 99.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{-24} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{+297}\right):\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	58.4%
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-75} \lor \neg \left(z \leq -1.1 \cdot 10^{-167}\right) \land z \leq 9.2 \cdot 10^{-140}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	58.7%
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-34}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-76} \lor \neg \left(z \leq -3.65 \cdot 10^{-168}\right) \land z \leq 2.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	58.5%
Cost	848

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1.75 \cdot 10^{+223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+134}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	58.6%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+223}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5
Accuracy	79.9%
Cost	448

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

Alternative 6
Accuracy	80.2%
Cost	448

\[x + \frac{x}{\frac{z}{y}} \]

Alternative 7
Accuracy	51.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023153 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

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