Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Accuracy: 80.5% → 97.4%

Time: 8.5s

Precision: binary64

Cost: 1480

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq 10^{+27}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]

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 (<= t_0 1e+27)
     (+ x (/ x (/ z y)))
     (if (<= t_0 2e+303) t_0 (* x (+ 1.0 (/ y z)))))))

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 <= 1e+27) {
		tmp = x + (x / (z / y));
	} else if (t_0 <= 2e+303) {
		tmp = t_0;
	} else {
		tmp = x * (1.0 + (y / z));
	}
	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 <= 1d+27) then
        tmp = x + (x / (z / y))
    else if (t_0 <= 2d+303) then
        tmp = t_0
    else
        tmp = x * (1.0d0 + (y / z))
    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 <= 1e+27) {
		tmp = x + (x / (z / y));
	} else if (t_0 <= 2e+303) {
		tmp = t_0;
	} else {
		tmp = x * (1.0 + (y / z));
	}
	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 <= 1e+27:
		tmp = x + (x / (z / y))
	elif t_0 <= 2e+303:
		tmp = t_0
	else:
		tmp = x * (1.0 + (y / z))
	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 <= 1e+27)
		tmp = Float64(x + Float64(x / Float64(z / y)));
	elseif (t_0 <= 2e+303)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(1.0 + Float64(y / z)));
	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 <= 1e+27)
		tmp = x + (x / (z / y));
	elseif (t_0 <= 2e+303)
		tmp = t_0;
	else
		tmp = x * (1.0 + (y / z));
	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[LessEqual[t$95$0, 1e+27], N[(x + N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+303], t$95$0, N[(x * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	80.5%
Target	95.4%
Herbie	97.4%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 1e27

   1. Initial program 83.6%

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

   2. Simplified 96.5%

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

      [Start] 83.6	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 79.1	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
      distribute-rgt-in [=>] 79.1	\[ \color{blue}{y \cdot \frac{x}{z} + z \cdot \frac{x}{z}} \]
      *-commutative [=>] 79.1	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{z} \cdot z} \]
      associate-/r/ [<=] 93.3	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{\frac{z}{z}}} \]
      *-inverses [=>] 93.3	\[ y \cdot \frac{x}{z} + \frac{x}{\color{blue}{1}} \]
      /-rgt-identity [=>] 93.3	\[ y \cdot \frac{x}{z} + \color{blue}{x} \]
      associate-*r/ [=>] 93.1	\[ \color{blue}{\frac{y \cdot x}{z}} + x \]
      *-commutative [<=] 93.1	\[ \frac{\color{blue}{x \cdot y}}{z} + x \]
      associate-*r/ [<=] 96.5	\[ \color{blue}{x \cdot \frac{y}{z}} + x \]
      fma-def [=>] 96.5	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]

   3. Applied egg-rr 96.6%

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

      [Start] 96.5	\[ \mathsf{fma}\left(x, \frac{y}{z}, x\right) \]
      fma-udef [=>] 96.5	\[ \color{blue}{x \cdot \frac{y}{z} + x} \]
      clear-num [=>] 96.4	\[ x \cdot \color{blue}{\frac{1}{\frac{z}{y}}} + x \]
      un-div-inv [=>] 96.6	\[ \color{blue}{\frac{x}{\frac{z}{y}}} + x \]

   if 1e27 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2e303

   1. Initial program 99.7%

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

   if 2e303 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 3.2%

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

   2. Simplified 99.2%

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

      [Start] 3.2	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 98.1	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
      distribute-rgt-in [=>] 98.1	\[ \color{blue}{y \cdot \frac{x}{z} + z \cdot \frac{x}{z}} \]
      *-commutative [=>] 98.1	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{z} \cdot z} \]
      associate-/r/ [<=] 98.2	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{\frac{z}{z}}} \]
      *-inverses [=>] 98.2	\[ y \cdot \frac{x}{z} + \frac{x}{\color{blue}{1}} \]
      /-rgt-identity [=>] 98.2	\[ y \cdot \frac{x}{z} + \color{blue}{x} \]
      associate-*r/ [=>] 64.7	\[ \color{blue}{\frac{y \cdot x}{z}} + x \]
      *-commutative [<=] 64.7	\[ \frac{\color{blue}{x \cdot y}}{z} + x \]
      associate-*r/ [<=] 99.2	\[ \color{blue}{x \cdot \frac{y}{z}} + x \]
      fma-def [=>] 99.2	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]

   3. Taylor expanded in x around 0 99.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 10^{+27}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	67.8%
Cost	1114

\[\begin{array}{l} \mathbf{if}\;z \leq -58:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-36} \lor \neg \left(z \leq -1.2 \cdot 10^{-45}\right) \land \left(z \leq 2.7 \cdot 10^{-108} \lor \neg \left(z \leq 2.05 \cdot 10^{-8}\right) \land z \leq 9.6 \cdot 10^{+43}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	67.8%
Cost	1113

\[\begin{array}{l} \mathbf{if}\;z \leq -34:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-107} \lor \neg \left(z \leq 2.2 \cdot 10^{-8}\right) \land z \leq 9.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	67.9%
Cost	1112

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -34:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	66.1%
Cost	1112

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -34:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	95.2%
Cost	448

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

Alternative 6
Accuracy	95.5%
Cost	448

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

Alternative 7
Accuracy	95.4%
Cost	448

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

Alternative 8
Accuracy	60.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(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))