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

Average Error: 16.4 → 5.0

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -6.116899373379197 \cdot 10^{-258}:\\ \;\;\;\;x + y \cdot \left(\frac{z}{t - a} + \left(1 - \frac{t}{t - a}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;x + \left(\frac{z}{\frac{t - a}{y}} - y \cdot \frac{a}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + \left(z - t\right) \cdot \frac{y}{t - a}\right)\\ \end{array} \]

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)))))
   (if (<= t_1 -6.116899373379197e-258)
     (+ x (* y (+ (/ z (- t a)) (- 1.0 (/ t (- t a))))))
     (if (<= t_1 0.0)
       (+ x (- (/ z (/ (- t a) y)) (* y (/ a t))))
       (+ x (+ y (* (- z t) (/ y (- t 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 tmp;
	if (t_1 <= -6.116899373379197e-258) {
		tmp = x + (y * ((z / (t - a)) + (1.0 - (t / (t - a)))));
	} else if (t_1 <= 0.0) {
		tmp = x + ((z / ((t - a) / y)) - (y * (a / t)));
	} else {
		tmp = x + (y + ((z - t) * (y / (t - 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) :: tmp
    t_1 = (x + y) - ((y * (z - t)) / (a - t))
    if (t_1 <= (-6.116899373379197d-258)) then
        tmp = x + (y * ((z / (t - a)) + (1.0d0 - (t / (t - a)))))
    else if (t_1 <= 0.0d0) then
        tmp = x + ((z / ((t - a) / y)) - (y * (a / t)))
    else
        tmp = x + (y + ((z - t) * (y / (t - 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 tmp;
	if (t_1 <= -6.116899373379197e-258) {
		tmp = x + (y * ((z / (t - a)) + (1.0 - (t / (t - a)))));
	} else if (t_1 <= 0.0) {
		tmp = x + ((z / ((t - a) / y)) - (y * (a / t)));
	} else {
		tmp = x + (y + ((z - t) * (y / (t - 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))
	tmp = 0
	if t_1 <= -6.116899373379197e-258:
		tmp = x + (y * ((z / (t - a)) + (1.0 - (t / (t - a)))))
	elif t_1 <= 0.0:
		tmp = x + ((z / ((t - a) / y)) - (y * (a / t)))
	else:
		tmp = x + (y + ((z - t) * (y / (t - 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)))
	tmp = 0.0
	if (t_1 <= -6.116899373379197e-258)
		tmp = Float64(x + Float64(y * Float64(Float64(z / Float64(t - a)) + Float64(1.0 - Float64(t / Float64(t - a))))));
	elseif (t_1 <= 0.0)
		tmp = Float64(x + Float64(Float64(z / Float64(Float64(t - a) / y)) - Float64(y * Float64(a / t))));
	else
		tmp = Float64(x + Float64(y + Float64(Float64(z - t) * Float64(y / Float64(t - 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));
	tmp = 0.0;
	if (t_1 <= -6.116899373379197e-258)
		tmp = x + (y * ((z / (t - a)) + (1.0 - (t / (t - a)))));
	elseif (t_1 <= 0.0)
		tmp = x + ((z / ((t - a) / y)) - (y * (a / t)));
	else
		tmp = x + (y + ((z - t) * (y / (t - 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]}, If[LessEqual[t$95$1, -6.116899373379197e-258], N[(x + N[(y * N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(x + N[(N[(z / N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - N[(y * N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y + N[(N[(z - t), $MachinePrecision] * N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision]), $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	8.9
Herbie	5.0

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -6.1168993733791966e-258

   1. Initial program 13.0

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

   2. Simplified 5.7

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

   3. Taylor expanded in y around 0 5.7

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

   4. Simplified 3.9

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

   if -6.1168993733791966e-258 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 59.0

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

   2. Simplified 35.7

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

   3. Taylor expanded in z around 0 35.9

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

   4. Simplified 31.7

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

   5. Applied egg-rr 31.7

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

   6. Taylor expanded in t around inf 5.9

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

   7. Simplified 5.8

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

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

   1. Initial program 12.4

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

   2. Simplified 5.5

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

   3. Taylor expanded in y around 0 5.5

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

   4. Simplified 3.6

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

   5. Taylor expanded in y around 0 5.5

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

   6. Simplified 5.9

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

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq -6.116899373379197 \cdot 10^{-258}:\\ \;\;\;\;x + y \cdot \left(\frac{z}{t - a} + \left(1 - \frac{t}{t - a}\right)\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{y \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;x + \left(\frac{z}{\frac{t - a}{y}} - y \cdot \frac{a}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y + \left(z - t\right) \cdot \frac{y}{t - a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))