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

Percentage Accurate: 89.0% → 98.6%

Time: 16.6s

Precision: binary64

Cost: 8964

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

↓

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

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)))
   (if (<= (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0)) INFINITY)
     (/ (+ x (fma y (/ z t_1) (/ (- x) t_1))) (+ x 1.0))
     (-
      (+ (/ y (* t (+ x 1.0))) (/ x (+ x 1.0)))
      (/ x (* (* z t) (+ x 1.0)))))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double tmp;
	if (((x + (((y * z) - x) / t_1)) / (x + 1.0)) <= ((double) INFINITY)) {
		tmp = (x + fma(y, (z / t_1), (-x / t_1))) / (x + 1.0);
	} else {
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((z * t) * (x + 1.0)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0)) <= Inf)
		tmp = Float64(Float64(x + fma(y, Float64(z / t_1), Float64(Float64(-x) / t_1))) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(y / Float64(t * Float64(x + 1.0))) + Float64(x / Float64(x + 1.0))) - Float64(x / Float64(Float64(z * t) * Float64(x + 1.0))));
	end
	return tmp
end

Wolfram

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

↓

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

Target

Original	89.0%
Target	99.4%
Herbie	98.6%

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.8%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Simplified 91.8%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
      Step-by-step derivation

      [Start] 91.8	\[ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
      *-commutative [=>] 91.8	\[ \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Applied egg-rr 98.7%

      \[\leadsto \frac{x + \color{blue}{\mathsf{fma}\left(\frac{y}{1}, \frac{z}{z \cdot t - x}, -\frac{x}{z \cdot t - x}\right)}}{x + 1} \]
      Step-by-step derivation

      [Start] 91.8	\[ \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \]
      div-sub [=>] 91.8	\[ \frac{x + \color{blue}{\left(\frac{y \cdot z}{z \cdot t - x} - \frac{x}{z \cdot t - x}\right)}}{x + 1} \]
      *-un-lft-identity [=>] 91.8	\[ \frac{x + \left(\frac{y \cdot z}{\color{blue}{1 \cdot \left(z \cdot t - x\right)}} - \frac{x}{z \cdot t - x}\right)}{x + 1} \]
      times-frac [=>] 98.7	\[ \frac{x + \left(\color{blue}{\frac{y}{1} \cdot \frac{z}{z \cdot t - x}} - \frac{x}{z \cdot t - x}\right)}{x + 1} \]
      fma-neg [=>] 98.7	\[ \frac{x + \color{blue}{\mathsf{fma}\left(\frac{y}{1}, \frac{z}{z \cdot t - x}, -\frac{x}{z \cdot t - x}\right)}}{x + 1} \]

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

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
      Step-by-step derivation

      [Start] 0.0	\[ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
      *-commutative [=>] 0.0	\[ \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Taylor expanded in t around inf 100.0%

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

4. Final simplification 98.8%

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

Alternatives

Alternative 1
Accuracy	94.5%
Cost	2628

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{+272}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{\left(z \cdot t\right) \cdot \left(x + 1\right)}\\ \end{array} \]

Alternative 2
Accuracy	94.4%
Cost	2244

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{+272}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 3
Accuracy	67.4%
Cost	1112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-121}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 0.01:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	79.8%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -260000000000 \lor \neg \left(t \leq 5.4 \cdot 10^{-203}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - \frac{y}{\frac{x}{z}}\right)}{x + 1}\\ \end{array} \]

Alternative 5
Accuracy	67.5%
Cost	981

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-120} \lor \neg \left(x \leq 1.35 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]

Alternative 6
Accuracy	79.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-60} \lor \neg \left(t \leq 5.5 \cdot 10^{-196}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{z}{x + 1}\\ \end{array} \]

Alternative 7
Accuracy	80.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+15} \lor \neg \left(t \leq 5.5 \cdot 10^{-196}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\frac{x}{z}}}{x + 1}\\ \end{array} \]

Alternative 8
Accuracy	67.3%
Cost	856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 0.01:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	77.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-32} \lor \neg \left(x \leq 190000\right):\\ \;\;\;\;1 - \frac{y}{x \cdot \frac{x}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 10
Accuracy	67.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-33}:\\ \;\;\;\;1 - \frac{y}{x \cdot \frac{x}{z}}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 11
Accuracy	55.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-99}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.01:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	53.4%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023163 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

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