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

Percentage Accurate: 89.1% → 95.6%

Time: 10.5s

Precision: binary64

Cost: 3400

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{t}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+223}:\\ \;\;\;\;\frac{t_1}{x + 1}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - \frac{x}{z \cdot t}}{x + 1}\\ \end{array} \end{array} \]

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 (+ x (/ y t)))
        (t_2 (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))))
   (if (<= t_2 -5e+223)
     (/ t_1 (+ x 1.0))
     (if (<= t_2 2e+292) t_2 (/ (- t_1 (/ 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 = x + (y / t);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+223) {
		tmp = t_1 / (x + 1.0);
	} else if (t_2 <= 2e+292) {
		tmp = t_2;
	} else {
		tmp = (t_1 - (x / (z * t))) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x + (y / t)
    t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0d0)
    if (t_2 <= (-5d+223)) then
        tmp = t_1 / (x + 1.0d0)
    else if (t_2 <= 2d+292) then
        tmp = t_2
    else
        tmp = (t_1 - (x / (z * t))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (y / t);
	double t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+223) {
		tmp = t_1 / (x + 1.0);
	} else if (t_2 <= 2e+292) {
		tmp = t_2;
	} else {
		tmp = (t_1 - (x / (z * t))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

↓

def code(x, y, z, t):
	t_1 = x + (y / t)
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)
	tmp = 0
	if t_2 <= -5e+223:
		tmp = t_1 / (x + 1.0)
	elif t_2 <= 2e+292:
		tmp = t_2
	else:
		tmp = (t_1 - (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(x + Float64(y / t))
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e+223)
		tmp = Float64(t_1 / Float64(x + 1.0));
	elseif (t_2 <= 2e+292)
		tmp = t_2;
	else
		tmp = Float64(Float64(t_1 - Float64(x / Float64(z * t))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / t);
	t_2 = (x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -5e+223)
		tmp = t_1 / (x + 1.0);
	elseif (t_2 <= 2e+292)
		tmp = t_2;
	else
		tmp = (t_1 - (x / (z * t))) / (x + 1.0);
	end
	tmp_2 = 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[(x + N[(y / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+223], N[(t$95$1 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+292], t$95$2, N[(N[(t$95$1 - N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	89.1%
Target	99.4%
Herbie	95.6%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -4.99999999999999985e223

   1. Initial program 48.0%

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

   2. Simplified 48.0%

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

      [Start] 48.0%	\[ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
      *-commutative [=>] 48.0%	\[ \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Taylor expanded in z around inf 84.9%

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

   if -4.99999999999999985e223 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 2e292

   1. Initial program 99.8%

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

   if 2e292 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 20.0%

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

   2. Simplified 20.0%

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

      [Start] 20.0%	\[ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
      *-commutative [=>] 20.0%	\[ \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Applied egg-rr 20.0%

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

      [Start] 20.0%	\[ \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \]
      fma-neg [=>] 20.0%	\[ \frac{x + \frac{y \cdot z - x}{\color{blue}{\mathsf{fma}\left(z, t, -x\right)}}}{x + 1} \]

   4. Taylor expanded in t around inf 86.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5 \cdot 10^{+223}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + \frac{y}{t}\right) - \frac{x}{z \cdot t}}{x + 1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.6%
Cost	3400

\[\begin{array}{l} t_1 := x + \frac{y}{t}\\ t_2 := \frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+223}:\\ \;\;\;\;\frac{t_1}{x + 1}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - \frac{x}{z \cdot t}}{x + 1}\\ \end{array} \]

Alternative 2
Accuracy	77.4%
Cost	1104

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;x \leq -15000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-108}:\\ \;\;\;\;y \cdot z + \left(1 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 0.00086:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\frac{x \cdot x}{z}}\\ \end{array} \]

Alternative 3
Accuracy	67.1%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-186}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x}}\\ \end{array} \]

Alternative 4
Accuracy	82.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.225 \lor \neg \left(t \leq 4.8 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x + 1}\\ \end{array} \]

Alternative 5
Accuracy	63.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -10 \lor \neg \left(t \leq 7.8 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 6
Accuracy	77.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -230000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 7
Accuracy	77.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -255000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\frac{x \cdot x}{z}}\\ \end{array} \]

Alternative 8
Accuracy	67.9%
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-52}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	67.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x}}\\ \end{array} \]

Alternative 10
Accuracy	67.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-74} \lor \neg \left(x \leq 4.7 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t}\\ \end{array} \]

Alternative 11
Accuracy	67.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-51}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-126}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	53.0%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023167 
(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)))