Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 10.0 → 0.1

Time: 14.0s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{elif}\;z \leq 100000000:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1}{z} \cdot \left(1 + y\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5e-25)
   (* x (+ -1.0 (/ (+ 1.0 y) z)))
   (if (<= z 100000000.0)
     (* (+ (- y z) 1.0) (/ x z))
     (* x (+ -1.0 (* (/ 1.0 z) (+ 1.0 y)))))))

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e-25) {
		tmp = x * (-1.0 + ((1.0 + y) / z));
	} else if (z <= 100000000.0) {
		tmp = ((y - z) + 1.0) * (x / z);
	} else {
		tmp = x * (-1.0 + ((1.0 / z) * (1.0 + y)));
	}
	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) + 1.0d0)) / 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) :: tmp
    if (z <= (-5.5d-25)) then
        tmp = x * ((-1.0d0) + ((1.0d0 + y) / z))
    else if (z <= 100000000.0d0) then
        tmp = ((y - z) + 1.0d0) * (x / z)
    else
        tmp = x * ((-1.0d0) + ((1.0d0 / z) * (1.0d0 + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e-25) {
		tmp = x * (-1.0 + ((1.0 + y) / z));
	} else if (z <= 100000000.0) {
		tmp = ((y - z) + 1.0) * (x / z);
	} else {
		tmp = x * (-1.0 + ((1.0 / z) * (1.0 + y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

↓

def code(x, y, z):
	tmp = 0
	if z <= -5.5e-25:
		tmp = x * (-1.0 + ((1.0 + y) / z))
	elif z <= 100000000.0:
		tmp = ((y - z) + 1.0) * (x / z)
	else:
		tmp = x * (-1.0 + ((1.0 / z) * (1.0 + y)))
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end

↓

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5e-25)
		tmp = Float64(x * Float64(-1.0 + Float64(Float64(1.0 + y) / z)));
	elseif (z <= 100000000.0)
		tmp = Float64(Float64(Float64(y - z) + 1.0) * Float64(x / z));
	else
		tmp = Float64(x * Float64(-1.0 + Float64(Float64(1.0 / z) * Float64(1.0 + y))));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * ((y - z) + 1.0)) / z;
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.5e-25)
		tmp = x * (-1.0 + ((1.0 + y) / z));
	elseif (z <= 100000000.0)
		tmp = ((y - z) + 1.0) * (x / z);
	else
		tmp = x * (-1.0 + ((1.0 / z) * (1.0 + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[z, -5.5e-25], N[(x * N[(-1.0 + N[(N[(1.0 + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 100000000.0], N[(N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(-1.0 + N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	10.0
Target	0.4
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -5.50000000000000004e-25

   1. Initial program 15.5

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

   2. Simplified 0.2

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

      [Start] 15.5	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      rational.json-simplify-2 [=>] 15.5	\[ \frac{\color{blue}{\left(\left(y - z\right) + 1\right) \cdot x}}{z} \]
      rational.json-simplify-49 [=>] 0.2	\[ \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]

   3. Applied egg-rr 0.1

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

   4. Taylor expanded in z around 0 5.1

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

   5. Simplified 0.2

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

      [Start] 5.1	\[ -1 \cdot x + \frac{\left(1 + y\right) \cdot x}{z} \]
      rational.json-simplify-1 [=>] 5.1	\[ \color{blue}{\frac{\left(1 + y\right) \cdot x}{z} + -1 \cdot x} \]
      rational.json-simplify-49 [=>] 0.2	\[ \color{blue}{x \cdot \frac{1 + y}{z}} + -1 \cdot x \]
      rational.json-simplify-51 [=>] 0.2	\[ \color{blue}{x \cdot \left(-1 + \frac{1 + y}{z}\right)} \]

   if -5.50000000000000004e-25 < z < 1e8

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}} \]

   if 1e8 < z

   1. Initial program 17.1

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

   2. Simplified 0.1

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

      [Start] 17.1	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      rational.json-simplify-2 [=>] 17.1	\[ \frac{\color{blue}{\left(\left(y - z\right) + 1\right) \cdot x}}{z} \]
      rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]

   3. Applied egg-rr 0.1

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

   4. Taylor expanded in z around 0 6.3

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

   5. Simplified 0.1

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

      [Start] 6.3	\[ -1 \cdot x + \frac{\left(1 + y\right) \cdot x}{z} \]
      rational.json-simplify-1 [=>] 6.3	\[ \color{blue}{\frac{\left(1 + y\right) \cdot x}{z} + -1 \cdot x} \]
      rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{x \cdot \frac{1 + y}{z}} + -1 \cdot x \]
      rational.json-simplify-51 [=>] 0.1	\[ \color{blue}{x \cdot \left(-1 + \frac{1 + y}{z}\right)} \]

   6. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{elif}\;z \leq 100000000:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{1}{z} \cdot \left(1 + y\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	20.9
Cost	852

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+52}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Error	11.7
Cost	848

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+165}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	11.8
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+169}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.1
Cost	840

\[\begin{array}{l} t_0 := x \cdot \left(-1 + \frac{1 + y}{z}\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 500000000000:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	4.6
Cost	712

\[\begin{array}{l} t_0 := x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{if}\;y \leq -2300000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	19.6
Cost	588

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+54}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 7
Error	3.8
Cost	576

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

Alternative 8
Error	19.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9
Error	33.5
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023075 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))