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

Average Error: 22.0 → 0.0

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ t_1 := x + t_0\\ \mathbf{if}\;y \leq -12134.734181339038:\\ \;\;\;\;\left(t_1 + \frac{1 - x}{{y}^{3}}\right) - \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 12997.544702284325:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \frac{1}{{y}^{3}}\right) - \frac{t_0 + \left(\left(x + -1\right) + \frac{1}{y}\right)}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))) (t_1 (+ x t_0)))
   (if (<= y -12134.734181339038)
     (- (+ t_1 (/ (- 1.0 x) (pow y 3.0))) (+ (/ 1.0 (* y y)) (/ (+ x -1.0) y)))
     (if (<= y 12997.544702284325)
       (fma y (/ (+ x -1.0) (+ y 1.0)) 1.0)
       (-
        (+ t_1 (/ 1.0 (pow y 3.0)))
        (/ (+ t_0 (+ (+ x -1.0) (/ 1.0 y))) y))))))

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 = x / (y * y);
	double t_1 = x + t_0;
	double tmp;
	if (y <= -12134.734181339038) {
		tmp = (t_1 + ((1.0 - x) / pow(y, 3.0))) - ((1.0 / (y * y)) + ((x + -1.0) / y));
	} else if (y <= 12997.544702284325) {
		tmp = fma(y, ((x + -1.0) / (y + 1.0)), 1.0);
	} else {
		tmp = (t_1 + (1.0 / pow(y, 3.0))) - ((t_0 + ((x + -1.0) + (1.0 / y))) / y);
	}
	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(x / Float64(y * y))
	t_1 = Float64(x + t_0)
	tmp = 0.0
	if (y <= -12134.734181339038)
		tmp = Float64(Float64(t_1 + Float64(Float64(1.0 - x) / (y ^ 3.0))) - Float64(Float64(1.0 / Float64(y * y)) + Float64(Float64(x + -1.0) / y)));
	elseif (y <= 12997.544702284325)
		tmp = fma(y, Float64(Float64(x + -1.0) / Float64(y + 1.0)), 1.0);
	else
		tmp = Float64(Float64(t_1 + Float64(1.0 / (y ^ 3.0))) - Float64(Float64(t_0 + Float64(Float64(x + -1.0) + Float64(1.0 / y))) / y));
	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[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + t$95$0), $MachinePrecision]}, If[LessEqual[y, -12134.734181339038], N[(N[(t$95$1 + N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 12997.544702284325], N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$1 + N[(1.0 / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 + N[(N[(x + -1.0), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Target

Original	22.0
Target	0.3
Herbie	0.0

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

Derivation

1. Split input into 3 regimes

2. if y < -12134.73418133904

   1. Initial program 45.0

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

   2. Simplified 30.1

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

   3. Taylor expanded in y around inf 0.0

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

   4. Simplified 0.0

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

   5. Applied associate--r+_binary64 0.0

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

   6. Simplified 0.0

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

   if -12134.73418133904 < y < 12997.544702284325

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   if 12997.544702284325 < y

   1. Initial program 44.6

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

   2. Simplified 28.1

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

   3. Taylor expanded in y around inf 0.0

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

   4. Simplified 0.0

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

   5. Applied div-inv_binary64 0.0

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

   6. Applied *-un-lft-identity_binary64 0.0

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

   7. Applied times-frac_binary64 0.0

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

   8. Applied distribute-rgt-out_binary64 0.0

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

   9. Applied cube-mult_binary64 0.0

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

   10. Applied *-un-lft-identity_binary64 0.0

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

   11. Applied times-frac_binary64 0.0

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

   12. Applied distribute-lft-out_binary64 0.0

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

   13. Applied cancel-sign-sub-inv_binary64 0.0

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

   14. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12134.734181339038:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1 - x}{{y}^{3}}\right) - \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 12997.544702284325:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \frac{\frac{x}{y \cdot y} + \left(\left(x + -1\right) + \frac{1}{y}\right)}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))