Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.0 → 0.4

Time: 2.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := x + \frac{x \cdot y}{z}\\ t_1 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (* x y) z))) (t_1 (/ (* x (+ y z)) z)))
   (if (<= t_1 (- INFINITY))
     (/ x (/ z (+ y z)))
     (if (<= t_1 -1e+80)
       t_0
       (if (<= t_1 5e-38)
         (fma x (/ y z) x)
         (if (<= t_1 5e+299) t_0 (fma y (/ x z) x)))))))

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 + ((x * y) / z);
	double t_1 = (x * (y + z)) / z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x / (z / (y + z));
	} else if (t_1 <= -1e+80) {
		tmp = t_0;
	} else if (t_1 <= 5e-38) {
		tmp = fma(x, (y / z), x);
	} else if (t_1 <= 5e+299) {
		tmp = t_0;
	} else {
		tmp = fma(y, (x / z), x);
	}
	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(x * y) / z))
	t_1 = Float64(Float64(x * Float64(y + z)) / z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x / Float64(z / Float64(y + z)));
	elseif (t_1 <= -1e+80)
		tmp = t_0;
	elseif (t_1 <= 5e-38)
		tmp = fma(x, Float64(y / z), x);
	elseif (t_1 <= 5e+299)
		tmp = t_0;
	else
		tmp = fma(y, Float64(x / z), x);
	end
	return 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[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x / N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+80], t$95$0, If[LessEqual[t$95$1, 5e-38], N[(x * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+299], t$95$0, N[(y * N[(x / z), $MachinePrecision] + x), $MachinePrecision]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.0
Target	2.9
Herbie	0.4

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{z}{y + z}}} \]

   3. Applied egg-rr 0.0

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -1e80 or 5.00000000000000033e-38 < (/.f64 (*.f64 x (+.f64 y z)) z) < 5.0000000000000003e299

   1. Initial program 0.2

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

   2. Applied egg-rr 7.4

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{z}{y + z}}} \]

   3. Taylor expanded in z around 0 0.2

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

   if -1e80 < (/.f64 (*.f64 x (+.f64 y z)) z) < 5.00000000000000033e-38

   1. Initial program 5.8

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

   2. Simplified 0.5

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

   if 5.0000000000000003e299 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 60.1

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

   2. Simplified 1.6

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

   3. Taylor expanded in x around 0 1.6

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

   4. Simplified 1.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z}, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1 \cdot 10^{+80}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+299}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))