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

Average Error: 10.7 → 0.3

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y}{z} + \frac{1}{z}\\ t_1 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \left(\sqrt[3]{t_0} \cdot \sqrt[3]{{t_0}^{2}} + -1\right)\\ \mathbf{elif}\;t_1 \leq 6 \cdot 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \end{array} \]

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 z))) (t_1 (/ (* x (+ (- y z) 1.0)) z)))
   (if (<= t_1 (- INFINITY))
     (* x (+ (* (cbrt t_0) (cbrt (pow t_0 2.0))) -1.0))
     (if (<= t_1 6e+304) (- (/ (fma x y x) z) x) (- (* y (/ x z)) x)))))

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 / z);
	double t_1 = (x * ((y - z) + 1.0)) / z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x * ((cbrt(t_0) * cbrt(pow(t_0, 2.0))) + -1.0);
	} else if (t_1 <= 6e+304) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = (y * (x / z)) - x;
	}
	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) + Float64(1.0 / z))
	t_1 = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x * Float64(Float64(cbrt(t_0) * cbrt((t_0 ^ 2.0))) + -1.0));
	elseif (t_1 <= 6e+304)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = Float64(Float64(y * Float64(x / z)) - x);
	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] + N[(1.0 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x * N[(N[(N[Power[t$95$0, 1/3], $MachinePrecision] * N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 6e+304], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.7
Target	0.5
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 20.6

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

   3. Taylor expanded in x around 0 0.0

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

   4. Applied egg-rr 2.3

      \[\leadsto \left(\color{blue}{\sqrt[3]{\frac{y}{z} + \frac{1}{z}} \cdot \sqrt[3]{{\left(\frac{y}{z} + \frac{1}{z}\right)}^{2}}} - 1\right) \cdot x \]

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 5.9999999999999996e304

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   if 5.9999999999999996e304 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 62.5

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

   2. Simplified 21.6

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

   3. Taylor expanded in y around inf 22.4

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

   4. Simplified 0.8

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} - x \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -\infty:\\ \;\;\;\;x \cdot \left(\sqrt[3]{\frac{y}{z} + \frac{1}{z}} \cdot \sqrt[3]{{\left(\frac{y}{z} + \frac{1}{z}\right)}^{2}} + -1\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 6 \cdot 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))