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

Average Accuracy: 83.3% → 99.2%

Time: 7.5s

Precision: binary64

Cost: 8136

Math

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

↓

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)) (t_1 (/ (* x t_0) z)))
   (if (<= t_1 (- INFINITY))
     (- (* x (/ y z)) x)
     (if (<= t_1 5e+263) (- (/ (fma x y x) z) x) (/ x (/ z t_0))))))

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 t_0 = (y - z) + 1.0;
	double t_1 = (x * t_0) / z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x * (y / z)) - x;
	} else if (t_1 <= 5e+263) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = x / (z / t_0);
	}
	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)
	t_0 = Float64(Float64(y - z) + 1.0)
	t_1 = Float64(Float64(x * t_0) / z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x * Float64(y / z)) - x);
	elseif (t_1 <= 5e+263)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = Float64(x / Float64(z / t_0));
	end
	return 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_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[t$95$1, 5e+263], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	83.3%
Target	99.4%
Herbie	99.2%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -inf.0

   1. Initial program 0.0%

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

   2. Simplified 69.7%

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

      [Start] 0.0	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      +-commutative [=>] 0.0	\[ \frac{x \cdot \color{blue}{\left(1 + \left(y - z\right)\right)}}{z} \]
      sub-neg [=>] 0.0	\[ \frac{x \cdot \left(1 + \color{blue}{\left(y + \left(-z\right)\right)}\right)}{z} \]
      +-commutative [=>] 0.0	\[ \frac{x \cdot \left(1 + \color{blue}{\left(\left(-z\right) + y\right)}\right)}{z} \]
      associate-+r+ [=>] 0.0	\[ \frac{x \cdot \color{blue}{\left(\left(1 + \left(-z\right)\right) + y\right)}}{z} \]
      unsub-neg [=>] 0.0	\[ \frac{x \cdot \left(\color{blue}{\left(1 - z\right)} + y\right)}{z} \]
      associate-+l- [=>] 0.0	\[ \frac{x \cdot \color{blue}{\left(1 - \left(z - y\right)\right)}}{z} \]
      distribute-lft-out-- [<=] 0.0	\[ \frac{\color{blue}{x \cdot 1 - x \cdot \left(z - y\right)}}{z} \]
      *-rgt-identity [=>] 0.0	\[ \frac{\color{blue}{x} - x \cdot \left(z - y\right)}{z} \]
      distribute-rgt-out-- [<=] 0.0	\[ \frac{x - \color{blue}{\left(z \cdot x - y \cdot x\right)}}{z} \]
      sub-neg [=>] 0.0	\[ \frac{x - \color{blue}{\left(z \cdot x + \left(-y \cdot x\right)\right)}}{z} \]
      +-commutative [=>] 0.0	\[ \frac{x - \color{blue}{\left(\left(-y \cdot x\right) + z \cdot x\right)}}{z} \]
      associate--r+ [=>] 0.0	\[ \frac{\color{blue}{\left(x - \left(-y \cdot x\right)\right) - z \cdot x}}{z} \]
      div-sub [=>] 0.0	\[ \color{blue}{\frac{x - \left(-y \cdot x\right)}{z} - \frac{z \cdot x}{z}} \]

   3. Taylor expanded in y around inf 69.7%

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

   4. Simplified 99.9%

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

      [Start] 69.7	\[ \frac{y \cdot x}{z} - x \]
      associate-/l* [=>] 99.9	\[ \color{blue}{\frac{y}{\frac{z}{x}}} - x \]
      associate-/r/ [=>] 99.9	\[ \color{blue}{\frac{y}{z} \cdot x} - x \]

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 5.00000000000000022e263

   1. Initial program 99.8%

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

   2. Simplified 99.9%

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

      [Start] 99.8	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      +-commutative [=>] 99.8	\[ \frac{x \cdot \color{blue}{\left(1 + \left(y - z\right)\right)}}{z} \]
      sub-neg [=>] 99.8	\[ \frac{x \cdot \left(1 + \color{blue}{\left(y + \left(-z\right)\right)}\right)}{z} \]
      +-commutative [=>] 99.8	\[ \frac{x \cdot \left(1 + \color{blue}{\left(\left(-z\right) + y\right)}\right)}{z} \]
      associate-+r+ [=>] 99.8	\[ \frac{x \cdot \color{blue}{\left(\left(1 + \left(-z\right)\right) + y\right)}}{z} \]
      unsub-neg [=>] 99.8	\[ \frac{x \cdot \left(\color{blue}{\left(1 - z\right)} + y\right)}{z} \]
      associate-+l- [=>] 99.8	\[ \frac{x \cdot \color{blue}{\left(1 - \left(z - y\right)\right)}}{z} \]
      distribute-lft-out-- [<=] 99.8	\[ \frac{\color{blue}{x \cdot 1 - x \cdot \left(z - y\right)}}{z} \]
      *-rgt-identity [=>] 99.8	\[ \frac{\color{blue}{x} - x \cdot \left(z - y\right)}{z} \]
      distribute-rgt-out-- [<=] 99.8	\[ \frac{x - \color{blue}{\left(z \cdot x - y \cdot x\right)}}{z} \]
      sub-neg [=>] 99.8	\[ \frac{x - \color{blue}{\left(z \cdot x + \left(-y \cdot x\right)\right)}}{z} \]
      +-commutative [=>] 99.8	\[ \frac{x - \color{blue}{\left(\left(-y \cdot x\right) + z \cdot x\right)}}{z} \]
      associate--r+ [=>] 99.8	\[ \frac{\color{blue}{\left(x - \left(-y \cdot x\right)\right) - z \cdot x}}{z} \]
      div-sub [=>] 99.8	\[ \color{blue}{\frac{x - \left(-y \cdot x\right)}{z} - \frac{z \cdot x}{z}} \]

   if 5.00000000000000022e263 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 24.5%

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

   2. Simplified 94.2%

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

      [Start] 24.5	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 94.2	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	67.0%
Cost	980

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-217}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 155:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Accuracy	79.9%
Cost	849

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+100}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 29000000000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{+75} \lor \neg \left(y \leq 2.3 \cdot 10^{+115}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3
Accuracy	79.6%
Cost	849

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+102}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 7000000000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+77} \lor \neg \left(y \leq 4.2 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Accuracy	79.7%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 34000000000:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+77}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+116}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 5
Accuracy	99.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-20} \lor \neg \left(z \leq 2.9 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \end{array} \]

Alternative 6
Accuracy	99.8%
Cost	840

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \end{array} \]

Alternative 7
Accuracy	93.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -175000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8
Accuracy	98.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.96 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \end{array} \]

Alternative 9
Accuracy	93.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -175000:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} - x\\ \end{array} \]

Alternative 10
Accuracy	70.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 11
Accuracy	49.1%
Cost	128

\[-x \]

Alternative 12
Accuracy	3.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(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))