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

Average Error: 6.9 → 2.0

Time: 13.6s

Precision: binary64

Cost: 3400

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -5000:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{t_1}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+281}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \]

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)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -5000.0)
     (* (/ y (+ x 1.0)) (/ z t_1))
     (if (<= t_2 4e+281) t_2 (+ (/ x (+ x 1.0)) (/ y (* 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 t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5000.0) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 4e+281) {
		tmp = t_2;
	} else {
		tmp = (x / (x + 1.0)) + (y / (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 = (z * t) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_2 <= (-5000.0d0)) then
        tmp = (y / (x + 1.0d0)) * (z / t_1)
    else if (t_2 <= 4d+281) then
        tmp = t_2
    else
        tmp = (x / (x + 1.0d0)) + (y / (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 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5000.0) {
		tmp = (y / (x + 1.0)) * (z / t_1);
	} else if (t_2 <= 4e+281) {
		tmp = t_2;
	} else {
		tmp = (x / (x + 1.0)) + (y / (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 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -5000.0:
		tmp = (y / (x + 1.0)) * (z / t_1)
	elif t_2 <= 4e+281:
		tmp = t_2
	else:
		tmp = (x / (x + 1.0)) + (y / (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)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5000.0)
		tmp = Float64(Float64(y / Float64(x + 1.0)) * Float64(z / t_1));
	elseif (t_2 <= 4e+281)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(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 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -5000.0)
		tmp = (y / (x + 1.0)) * (z / t_1);
	elseif (t_2 <= 4e+281)
		tmp = t_2;
	else
		tmp = (x / (x + 1.0)) + (y / (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[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5000.0], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+281], t$95$2, N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	6.9
Target	0.3
Herbie	2.0

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

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)) < -5e3

   1. Initial program 17.5

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

   2. Simplified 17.5

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
      Proof
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z t) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 t z)) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 18.2

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

   4. Simplified 5.9

      \[\leadsto \color{blue}{\frac{y}{x + 1} \cdot \frac{z}{t \cdot z - x}} \]
      Proof
      (*.f64 (/.f64 y (+.f64 x 1)) (/.f64 z (-.f64 (*.f64 t z) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y (Rewrite=> +-commutative_binary64 (+.f64 1 x))) (/.f64 z (-.f64 (*.f64 t z) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y (+.f64 1 x)) (/.f64 z (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 z t)) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y (+.f64 1 x)) (/.f64 z (Rewrite=> fma-neg_binary64 (fma.f64 z t (neg.f64 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y z) (*.f64 (+.f64 1 x) (fma.f64 z t (neg.f64 x))))): 68 points increase in error, 29 points decrease in error
      (/.f64 (*.f64 y z) (*.f64 (+.f64 1 x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 z t) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 y z) (*.f64 (+.f64 1 x) (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 t z)) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 y z) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 t z) x) (+.f64 1 x)))): 0 points increase in error, 0 points decrease in error

   if -5e3 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 4.0000000000000001e281

   1. Initial program 0.7

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

   if 4.0000000000000001e281 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 61.2

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

   2. Simplified 61.2

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}} \]
      Proof
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z t) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 t z)) x))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in z around inf 10.1

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

   4. Taylor expanded in y around 0 9.5

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

   5. Simplified 9.5

      \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}} \]
      Proof
      (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 y (*.f64 t (+.f64 x 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 x))) (/.f64 y (*.f64 t (+.f64 x 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 1 x)) (/.f64 y (*.f64 t (Rewrite<= +-commutative_binary64 (+.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 1 x)) (/.f64 y (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 1 x) t)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5000:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{z}{z \cdot t - x}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 4 \cdot 10^{+281}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.8
Cost	1104

\[\begin{array}{l} t_1 := 1 - \frac{z}{x} \cdot \frac{y}{x}\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	12.0
Cost	1096

\[\begin{array}{l} t_1 := \frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;1 - \frac{\frac{y \cdot z}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	11.1
Cost	1096

\[\begin{array}{l} t_1 := \frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{if}\;t \leq -6.1 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{1 + \left(x - \frac{y}{\frac{x}{z}}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	20.4
Cost	972

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \left(1 + \frac{-1}{z \cdot t}\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	12.1
Cost	968

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-103}:\\ \;\;\;\;1 - \frac{\frac{y \cdot z}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.0
Cost	840

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-143}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	20.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-107}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-70}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	20.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	27.7
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{-301}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	28.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022331 
(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)))