Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.6 → 9.2

Time: 6.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -5.377347847996125 \cdot 10^{+110}:\\ \;\;\;\;\frac{y}{z} \cdot x + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.008333854918147 \cdot 10^{+137}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(\frac{z \cdot t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + \left(\frac{x}{\frac{z}{y}} + \frac{a}{\frac{z}{t}}\right)\right) - \left(\frac{y}{\frac{z}{t}} + \frac{x}{\frac{z}{a}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.377347847996125e+110)
   (+ (* (/ y z) x) (+ t (- (* t (/ a z)) (+ (/ t (/ z y)) (* x (/ a z))))))
   (if (<= z 2.008333854918147e+137)
     (+
      (* x (/ z (- a z)))
      (-
       (+ x (/ t (/ (- a z) y)))
       (+ (/ (* z t) (- a z)) (* x (/ y (- a z))))))
     (-
      (+ t (+ (/ x (/ z y)) (/ a (/ z t))))
      (+ (/ y (/ z t)) (/ x (/ z a)))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.377347847996125e+110) {
		tmp = ((y / z) * x) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	} else if (z <= 2.008333854918147e+137) {
		tmp = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - (((z * t) / (a - z)) + (x * (y / (a - z)))));
	} else {
		tmp = (t + ((x / (z / y)) + (a / (z / t)))) - ((y / (z / t)) + (x / (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 - x)) / (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 (z <= (-5.377347847996125d+110)) then
        tmp = ((y / z) * x) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))))
    else if (z <= 2.008333854918147d+137) then
        tmp = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - (((z * t) / (a - z)) + (x * (y / (a - z)))))
    else
        tmp = (t + ((x / (z / y)) + (a / (z / t)))) - ((y / (z / t)) + (x / (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 - x)) / (a - z));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.377347847996125e+110) {
		tmp = ((y / z) * x) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	} else if (z <= 2.008333854918147e+137) {
		tmp = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - (((z * t) / (a - z)) + (x * (y / (a - z)))));
	} else {
		tmp = (t + ((x / (z / y)) + (a / (z / t)))) - ((y / (z / t)) + (x / (z / a)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.377347847996125e+110:
		tmp = ((y / z) * x) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))))
	elif z <= 2.008333854918147e+137:
		tmp = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - (((z * t) / (a - z)) + (x * (y / (a - z)))))
	else:
		tmp = (t + ((x / (z / y)) + (a / (z / t)))) - ((y / (z / t)) + (x / (z / a)))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.377347847996125e+110)
		tmp = Float64(Float64(Float64(y / z) * x) + Float64(t + Float64(Float64(t * Float64(a / z)) - Float64(Float64(t / Float64(z / y)) + Float64(x * Float64(a / z))))));
	elseif (z <= 2.008333854918147e+137)
		tmp = Float64(Float64(x * Float64(z / Float64(a - z))) + Float64(Float64(x + Float64(t / Float64(Float64(a - z) / y))) - Float64(Float64(Float64(z * t) / Float64(a - z)) + Float64(x * Float64(y / Float64(a - z))))));
	else
		tmp = Float64(Float64(t + Float64(Float64(x / Float64(z / y)) + Float64(a / Float64(z / t)))) - Float64(Float64(y / Float64(z / t)) + Float64(x / Float64(z / a))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.377347847996125e+110)
		tmp = ((y / z) * x) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	elseif (z <= 2.008333854918147e+137)
		tmp = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - (((z * t) / (a - z)) + (x * (y / (a - z)))));
	else
		tmp = (t + ((x / (z / y)) + (a / (z / t)))) - ((y / (z / t)) + (x / (z / a)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.377347847996125e+110], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] + N[(t + N[(N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision] - N[(N[(t / N[(z / y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.008333854918147e+137], N[(N[(x * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + N[(t / N[(N[(a - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t + N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] + N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[(z / t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(z / a), $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	24.6
Target	12.2
Herbie	9.2

\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -5.37734784799612468e110

   1. Initial program 44.4

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

   2. Simplified 24.3

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

   3. Taylor expanded in z around inf 24.6

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

   4. Simplified 10.9

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

   if -5.37734784799612468e110 < z < 2.0083338549181468e137

   1. Initial program 13.8

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

   2. Simplified 9.2

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

   3. Taylor expanded in y around 0 11.1

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

   4. Simplified 8.6

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

   5. Taylor expanded in t around 0 8.7

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

   if 2.0083338549181468e137 < z

   1. Initial program 45.7

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

   2. Simplified 27.1

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

   3. Applied egg-rr 27.8

      \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{{\left(\sqrt[3]{t - x}\right)}^{2}}{{\left(\sqrt[3]{a - z}\right)}^{2}} \cdot \sqrt[3]{\frac{t - x}{a - z}}}, x\right) \]

   4. Taylor expanded in z around inf 23.7

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

   5. Simplified 9.3

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.377347847996125 \cdot 10^{+110}:\\ \;\;\;\;\frac{y}{z} \cdot x + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.008333854918147 \cdot 10^{+137}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(\frac{z \cdot t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + \left(\frac{x}{\frac{z}{y}} + \frac{a}{\frac{z}{t}}\right)\right) - \left(\frac{y}{\frac{z}{t}} + \frac{x}{\frac{z}{a}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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