Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.3 → 2.2

Time: 2.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ t_1 := \mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1.8899164894020197 \cdot 10^{+138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 1.4901022506184445 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma y (/ x z) x)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -1.8899164894020197e+138)
       t_0
       (if (<= t_0 1.4901022506184445e+44) (fma x (/ y z) x) t_1)))))

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 = fma(y, (x / z), x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -1.8899164894020197e+138) {
		tmp = t_0;
	} else if (t_0 <= 1.4901022506184445e+44) {
		tmp = fma(x, (y / z), x);
	} else {
		tmp = t_1;
	}
	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(Float64(x * Float64(y + z)) / z)
	t_1 = fma(y, Float64(x / z), x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -1.8899164894020197e+138)
		tmp = t_0;
	elseif (t_0 <= 1.4901022506184445e+44)
		tmp = fma(x, Float64(y / z), x);
	else
		tmp = t_1;
	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[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(x / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -1.8899164894020197e+138], t$95$0, If[LessEqual[t$95$0, 1.4901022506184445e+44], N[(x * N[(y / z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.3
Target	3.0
Herbie	2.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -inf.0 or 1.4901022506184445e44 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 28.9

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

   2. Simplified 5.2

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

   3. Taylor expanded in x around 0 5.2

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

   4. Simplified 5.2

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -1.8899164894020197e138

   1. Initial program 0.2

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

   if -1.8899164894020197e138 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.4901022506184445e44

   1. Initial program 5.1

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

   2. Simplified 0.9

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1.8899164894020197 \cdot 10^{+138}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 1.4901022506184445 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))