Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Average Error: 22.3 → 0.2

Time: 2.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ t_1 := \mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \mathbf{if}\;t_0 \leq 0.05:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;x + \left(1 + \frac{-1}{y}\right) \cdot \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y)))
        (t_1 (fma (- 1.0 x) (/ y (- -1.0 y)) 1.0)))
   (if (<= t_0 0.05)
     t_1
     (if (<= t_0 2.0) (+ x (* (+ 1.0 (/ -1.0 y)) (/ (- 1.0 x) y))) t_1))))

C

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

↓

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double t_1 = fma((1.0 - x), (y / (-1.0 - y)), 1.0);
	double tmp;
	if (t_0 <= 0.05) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x + ((1.0 + (-1.0 / y)) * ((1.0 - x) / y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

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

↓

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	t_1 = fma(Float64(1.0 - x), Float64(y / Float64(-1.0 - y)), 1.0)
	tmp = 0.0
	if (t_0 <= 0.05)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(x + Float64(Float64(1.0 + Float64(-1.0 / y)) * Float64(Float64(1.0 - x) / y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.05], t$95$1, If[LessEqual[t$95$0, 2.0], N[(x + N[(N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	22.3
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.050000000000000003 or 2 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 10.7

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)} \]

   if 0.050000000000000003 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 2

   1. Initial program 57.6

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

   2. Simplified 57.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)} \]

   3. Taylor expanded in y around inf 32.6

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

   4. Simplified 0.9

      \[\leadsto \color{blue}{x + \left(\frac{-1}{y} + 1\right) \cdot \frac{1 - x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 2:\\ \;\;\;\;x + \left(1 + \frac{-1}{y}\right) \cdot \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

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