Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.4 → 8.3

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t}\\ t_2 := 1 + \frac{a}{t}\\ \mathbf{if}\;t_1 \leq -9.590140231623633 \cdot 10^{-153}:\\ \;\;\;\;y + \left(x - \frac{y}{\frac{a - t}{z}}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(x + t_2 \cdot \frac{y \cdot z}{t}\right) - y \cdot \left(\frac{a}{t} \cdot t_2\right)\\ \mathbf{elif}\;t_1 \leq 1.5506509821356148 \cdot 10^{+260}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (/ (* y (- z t)) (- a t)))) (t_2 (+ 1.0 (/ a t))))
   (if (<= t_1 -9.590140231623633e-153)
     (+ y (- x (/ y (/ (- a t) z))))
     (if (<= t_1 0.0)
       (- (+ x (* t_2 (/ (* y z) t))) (* y (* (/ a t) t_2)))
       (if (<= t_1 1.5506509821356148e+260) t_1 (+ x (* (/ y t) (- z a))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x + y) - ((y * (z - t)) / (a - t));
	double t_2 = 1.0 + (a / t);
	double tmp;
	if (t_1 <= -9.590140231623633e-153) {
		tmp = y + (x - (y / ((a - t) / z)));
	} else if (t_1 <= 0.0) {
		tmp = (x + (t_2 * ((y * z) / t))) - (y * ((a / t) * t_2));
	} else if (t_1 <= 1.5506509821356148e+260) {
		tmp = t_1;
	} else {
		tmp = x + ((y / t) * (z - a));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + y) - ((y * (z - t)) / (a - t))
    t_2 = 1.0d0 + (a / t)
    if (t_1 <= (-9.590140231623633d-153)) then
        tmp = y + (x - (y / ((a - t) / z)))
    else if (t_1 <= 0.0d0) then
        tmp = (x + (t_2 * ((y * z) / t))) - (y * ((a / t) * t_2))
    else if (t_1 <= 1.5506509821356148d+260) then
        tmp = t_1
    else
        tmp = x + ((y / t) * (z - a))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x + y) - ((y * (z - t)) / (a - t));
	double t_2 = 1.0 + (a / t);
	double tmp;
	if (t_1 <= -9.590140231623633e-153) {
		tmp = y + (x - (y / ((a - t) / z)));
	} else if (t_1 <= 0.0) {
		tmp = (x + (t_2 * ((y * z) / t))) - (y * ((a / t) * t_2));
	} else if (t_1 <= 1.5506509821356148e+260) {
		tmp = t_1;
	} else {
		tmp = x + ((y / t) * (z - a));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = (x + y) - ((y * (z - t)) / (a - t))
	t_2 = 1.0 + (a / t)
	tmp = 0
	if t_1 <= -9.590140231623633e-153:
		tmp = y + (x - (y / ((a - t) / z)))
	elif t_1 <= 0.0:
		tmp = (x + (t_2 * ((y * z) / t))) - (y * ((a / t) * t_2))
	elif t_1 <= 1.5506509821356148e+260:
		tmp = t_1
	else:
		tmp = x + ((y / t) * (z - a))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x + y) - Float64(Float64(y * Float64(z - t)) / Float64(a - t)))
	t_2 = Float64(1.0 + Float64(a / t))
	tmp = 0.0
	if (t_1 <= -9.590140231623633e-153)
		tmp = Float64(y + Float64(x - Float64(y / Float64(Float64(a - t) / z))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x + Float64(t_2 * Float64(Float64(y * z) / t))) - Float64(y * Float64(Float64(a / t) * t_2)));
	elseif (t_1 <= 1.5506509821356148e+260)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(Float64(y / t) * Float64(z - a)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x + y) - ((y * (z - t)) / (a - t));
	t_2 = 1.0 + (a / t);
	tmp = 0.0;
	if (t_1 <= -9.590140231623633e-153)
		tmp = y + (x - (y / ((a - t) / z)));
	elseif (t_1 <= 0.0)
		tmp = (x + (t_2 * ((y * z) / t))) - (y * ((a / t) * t_2));
	elseif (t_1 <= 1.5506509821356148e+260)
		tmp = t_1;
	else
		tmp = x + ((y / t) * (z - a));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(a / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -9.590140231623633e-153], N[(y + N[(x - N[(y / N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x + N[(t$95$2 * N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(N[(a / t), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5506509821356148e+260], t$95$1, N[(x + N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.4
Target	8.5
Herbie	8.3

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -9.5901402316236329e-153

   1. Initial program 13.1

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

   2. Simplified 8.1

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

   3. Taylor expanded in y around 0 12.9

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

   4. Simplified 8.1

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

   5. Taylor expanded in z around inf 11.9

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

   6. Simplified 11.9

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

   7. Applied associate-/l*_binary64 9.3

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

   if -9.5901402316236329e-153 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 47.8

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

   2. Simplified 47.5

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

   3. Taylor expanded in t around inf 13.6

      \[\leadsto \color{blue}{\left(\frac{y \cdot z}{t} + \left(\frac{a \cdot \left(y \cdot z\right)}{{t}^{2}} + x\right)\right) - \left(\frac{a \cdot y}{t} + \frac{{a}^{2} \cdot y}{{t}^{2}}\right)} \]

   4. Simplified 9.8

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

   if 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 1.55065098213561478e260

   1. Initial program 1.6

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

   if 1.55065098213561478e260 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 41.8

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

   2. Simplified 20.3

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

   3. Taylor expanded in t around inf 33.9

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

   4. Simplified 21.0

      \[\leadsto \color{blue}{x + \frac{y}{t} \cdot \left(z - a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -9.590140231623633 \cdot 10^{-153}:\\ \;\;\;\;y + \left(x - \frac{y}{\frac{a - t}{z}}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;\left(x + \left(1 + \frac{a}{t}\right) \cdot \frac{y \cdot z}{t}\right) - y \cdot \left(\frac{a}{t} \cdot \left(1 + \frac{a}{t}\right)\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 1.5506509821356148 \cdot 10^{+260}:\\ \;\;\;\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - a\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-7) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

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