Crypto.Random.Test:calculate from crypto-random-0.0.9

Average Error: 5.8 → 0.1

Time: 5.4s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	5.8
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 5.8

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{y}{z}, x\right)} \]
   Proof
   (fma.f64 y (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (/.f64 y z)) x)): 0 points increase in error, 1 points decrease in error
   (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y y) z)) x): 39 points increase in error, 7 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 y y) z))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	10.9
Cost	1096

\[\begin{array}{l} t_0 := \frac{y \cdot y}{z}\\ t_1 := \frac{y}{\frac{z}{y}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{+60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.9
Cost	1096

\[\begin{array}{l} t_0 := \frac{y \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+43}:\\ \;\;\;\;y \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq 10^{+60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	18.5
Cost	980

\[\begin{array}{l} t_0 := \frac{y \cdot y}{z}\\ \mathbf{if}\;y \leq -1.06 \cdot 10^{+147}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+56}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.913500775179816 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y z)
  :name "Crypto.Random.Test:calculate from crypto-random-0.0.9"
  :precision binary64

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

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