Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.8 → 3.3

Time: 8.1s

Precision: binary64

Cost: 976

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{y + z}{z}\\ t_1 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-89}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-260}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (+ y z) z))) (t_1 (/ (* x (+ y z)) z)))
   (if (<= x -6.5e-89)
     (+ x (* x (/ y z)))
     (if (<= x -3.5e-260)
       t_1
       (if (<= x 8.6e-258) t_0 (if (<= x 2.1e+81) t_1 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 t_1 = (x * (y + z)) / z;
	double tmp;
	if (x <= -6.5e-89) {
		tmp = x + (x * (y / z));
	} else if (x <= -3.5e-260) {
		tmp = t_1;
	} else if (x <= 8.6e-258) {
		tmp = t_0;
	} else if (x <= 2.1e+81) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * ((y + z) / z)
    t_1 = (x * (y + z)) / z
    if (x <= (-6.5d-89)) then
        tmp = x + (x * (y / z))
    else if (x <= (-3.5d-260)) then
        tmp = t_1
    else if (x <= 8.6d-258) then
        tmp = t_0
    else if (x <= 2.1d+81) then
        tmp = t_1
    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 t_1 = (x * (y + z)) / z;
	double tmp;
	if (x <= -6.5e-89) {
		tmp = x + (x * (y / z));
	} else if (x <= -3.5e-260) {
		tmp = t_1;
	} else if (x <= 8.6e-258) {
		tmp = t_0;
	} else if (x <= 2.1e+81) {
		tmp = t_1;
	} 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)
	t_1 = (x * (y + z)) / z
	tmp = 0
	if x <= -6.5e-89:
		tmp = x + (x * (y / z))
	elif x <= -3.5e-260:
		tmp = t_1
	elif x <= 8.6e-258:
		tmp = t_0
	elif x <= 2.1e+81:
		tmp = t_1
	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(x * Float64(Float64(y + z) / z))
	t_1 = Float64(Float64(x * Float64(y + z)) / z)
	tmp = 0.0
	if (x <= -6.5e-89)
		tmp = Float64(x + Float64(x * Float64(y / z)));
	elseif (x <= -3.5e-260)
		tmp = t_1;
	elseif (x <= 8.6e-258)
		tmp = t_0;
	elseif (x <= 2.1e+81)
		tmp = t_1;
	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);
	t_1 = (x * (y + z)) / z;
	tmp = 0.0;
	if (x <= -6.5e-89)
		tmp = x + (x * (y / z));
	elseif (x <= -3.5e-260)
		tmp = t_1;
	elseif (x <= 8.6e-258)
		tmp = t_0;
	elseif (x <= 2.1e+81)
		tmp = t_1;
	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[(x * N[(N[(y + z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -6.5e-89], N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-260], t$95$1, If[LessEqual[x, 8.6e-258], t$95$0, If[LessEqual[x, 2.1e+81], t$95$1, t$95$0]]]]]]

Target

Original	12.8
Target	3.2
Herbie	3.3

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

Derivation

1. Split input into 3 regimes

2. if x < -6.50000000000000034e-89

   1. Initial program 16.6

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

   2. Simplified 0.7

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

      [Start] 16.6	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      rational.json-simplify-2 [=>] 16.6	\[ \frac{\color{blue}{\left(y + z\right) \cdot x}}{z} \]
      rational.json-simplify-49 [=>] 0.7	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

   3. Taylor expanded in y around 0 5.5

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

   4. Simplified 0.7

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

      [Start] 5.5	\[ \frac{y \cdot x}{z} + x \]
      rational.json-simplify-1 [=>] 5.5	\[ \color{blue}{x + \frac{y \cdot x}{z}} \]
      rational.json-simplify-49 [=>] 0.7	\[ x + \color{blue}{x \cdot \frac{y}{z}} \]

   if -6.50000000000000034e-89 < x < -3.5e-260 or 8.5999999999999992e-258 < x < 2.0999999999999999e81

   1. Initial program 4.7

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

   if -3.5e-260 < x < 8.5999999999999992e-258 or 2.0999999999999999e81 < x

   1. Initial program 23.3

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

   2. Simplified 3.7

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

      [Start] 23.3	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      rational.json-simplify-2 [=>] 23.3	\[ \frac{\color{blue}{\left(y + z\right) \cdot x}}{z} \]
      rational.json-simplify-49 [=>] 3.7	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-89}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-260}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.6
Cost	848

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -9.4 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-112}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 820:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	20.9
Cost	848

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	21.0
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+191}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+118}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Error	20.8
Cost	848

\[\begin{array}{l} t_0 := \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+116}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 5
Error	4.4
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y + z}{z}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-283}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	3.3
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y + z}{z}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-271}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	3.3
Cost	712

\[\begin{array}{l} t_0 := x + x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-271}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	25.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(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))