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

Average Error: 10.5 → 0.9

Time: 5.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1923891012664429 \cdot 10^{+57}:\\ \;\;\;\;x + \frac{\frac{y - z}{a - z}}{\frac{1}{t}}\\ \mathbf{elif}\;t \leq 1.4771816920899563 \cdot 10^{-224}:\\ \;\;\;\;x + \frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.1923891012664429e+57)
   (+ x (/ (/ (- y z) (- a z)) (/ 1.0 t)))
   (if (<= t 1.4771816920899563e-224)
     (+ x (/ (* t (- y z)) (- a z)))
     (+ x (* t (- (/ y (- a z)) (/ z (- a z))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.1923891012664429e+57) {
		tmp = x + (((y - z) / (a - z)) / (1.0 / t));
	} else if (t <= 1.4771816920899563e-224) {
		tmp = x + ((t * (y - z)) / (a - z));
	} else {
		tmp = x + (t * ((y / (a - z)) - (z / (a - z))));
	}
	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) / (a - z))
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) :: tmp
    if (t <= (-1.1923891012664429d+57)) then
        tmp = x + (((y - z) / (a - z)) / (1.0d0 / t))
    else if (t <= 1.4771816920899563d-224) then
        tmp = x + ((t * (y - z)) / (a - z))
    else
        tmp = x + (t * ((y / (a - z)) - (z / (a - z))))
    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) / (a - z));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.1923891012664429e+57) {
		tmp = x + (((y - z) / (a - z)) / (1.0 / t));
	} else if (t <= 1.4771816920899563e-224) {
		tmp = x + ((t * (y - z)) / (a - z));
	} else {
		tmp = x + (t * ((y / (a - z)) - (z / (a - z))));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.1923891012664429e+57:
		tmp = x + (((y - z) / (a - z)) / (1.0 / t))
	elif t <= 1.4771816920899563e-224:
		tmp = x + ((t * (y - z)) / (a - z))
	else:
		tmp = x + (t * ((y / (a - z)) - (z / (a - z))))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.1923891012664429e+57)
		tmp = Float64(x + Float64(Float64(Float64(y - z) / Float64(a - z)) / Float64(1.0 / t)));
	elseif (t <= 1.4771816920899563e-224)
		tmp = Float64(x + Float64(Float64(t * Float64(y - z)) / Float64(a - z)));
	else
		tmp = Float64(x + Float64(t * Float64(Float64(y / Float64(a - z)) - Float64(z / Float64(a - z)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.1923891012664429e+57)
		tmp = x + (((y - z) / (a - z)) / (1.0 / t));
	elseif (t <= 1.4771816920899563e-224)
		tmp = x + ((t * (y - z)) / (a - z));
	else
		tmp = x + (t * ((y / (a - z)) - (z / (a - z))));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.1923891012664429e+57], N[(x + N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4771816920899563e-224], N[(x + N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t * N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a - z), $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	10.5
Target	0.6
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if t < -1.19238910126644287e57

   1. Initial program 26.1

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

   2. Simplified 3.1

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

   3. Taylor expanded in y around 0 26.1

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

   4. Simplified 0.7

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

   5. Applied egg-rr 26.1

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

   6. Applied egg-rr 2.9

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

   7. Applied egg-rr 0.8

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

   if -1.19238910126644287e57 < t < 1.47718169208995632e-224

   1. Initial program 0.6

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

   2. Simplified 2.9

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

   3. Taylor expanded in y around 0 0.6

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

   4. Simplified 2.0

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

   5. Applied egg-rr 0.6

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

   if 1.47718169208995632e-224 < t

   1. Initial program 12.8

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

   2. Simplified 3.1

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

   3. Taylor expanded in y around 0 12.8

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

   4. Simplified 1.2

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1923891012664429 \cdot 10^{+57}:\\ \;\;\;\;x + \frac{\frac{y - z}{a - z}}{\frac{1}{t}}\\ \mathbf{elif}\;t \leq 1.4771816920899563 \cdot 10^{-224}:\\ \;\;\;\;x + \frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \end{array} \]

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, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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