Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 2.2 → 2.1

Time: 10.4s

Precision: binary64

Cost: 836

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{+57}:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) -4e+57) (/ z (/ t (- y x))) (+ x (/ (- y x) (/ t z)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -4e+57) {
		tmp = z / (t / (y - x));
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	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 - x) * (z / t))
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) :: tmp
    if ((z / t) <= (-4d+57)) then
        tmp = z / (t / (y - x))
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -4e+57) {
		tmp = z / (t / (y - x));
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	tmp = 0
	if (z / t) <= -4e+57:
		tmp = z / (t / (y - x))
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= -4e+57)
		tmp = Float64(z / Float64(t / Float64(y - x)));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z / t) <= -4e+57)
		tmp = z / (t / (y - x));
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], -4e+57], N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.2
Target	2.4
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} < -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -4.00000000000000019e57

   1. Initial program 7.8

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

   2. Simplified 7.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]
      Proof
      (fma.f64 (-.f64 y x) (/.f64 z t) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y x) (/.f64 z t)) x)): 1 points increase in error, 1 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (-.f64 y x) (/.f64 z t)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in z around inf 7.3

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

   4. Simplified 7.3

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]
      Proof
      (*.f64 z (/.f64 (-.f64 y x) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 z (Rewrite=> div-sub_binary64 (-.f64 (/.f64 y t) (/.f64 x t)))): 2 points increase in error, 2 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 y t) (/.f64 x t)) z)): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 6.9

      \[\leadsto \color{blue}{\frac{z}{\frac{t}{y - x}}} \]

   if -4.00000000000000019e57 < (/.f64 z t)

   1. Initial program 1.5

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

   2. Applied egg-rr 1.5

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{+57}:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.4
Cost	1944

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\frac{z}{\frac{-t}{x}}\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+23}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 4 \cdot 10^{+70}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+145}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{-t}\\ \end{array} \]

Alternative 2
Error	22.7
Cost	1424

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\frac{z}{\frac{-t}{x}}\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+23}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{-t}\\ \end{array} \]

Alternative 3
Error	22.7
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\frac{z}{\frac{-t}{x}}\\ \mathbf{elif}\;\frac{z}{t} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 4
Error	14.2
Cost	968

\[\begin{array}{l} t_1 := \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	14.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{-44}:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]

Alternative 6
Error	6.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-19}:\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \mathbf{elif}\;\frac{z}{t} \leq 0.0005:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]

Alternative 7
Error	22.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 8
Error	23.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-33}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 9
Error	26.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.921412181442593 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.34650554689846 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	31.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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