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

Average Error: 25.1 → 20.3

Time: 19.9s

Precision: binary64

Cost: 1232

Math

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

↓

\[\begin{array}{l} t_1 := t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{if}\;z \leq -2.946311588664566 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.849437786437544 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{t \cdot \left(y - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ t (/ (- x t) (/ z (- y a))))))
   (if (<= z -2.946311588664566e-7)
     t_1
     (if (<= z -5.2e-62)
       (* (/ t (- a z)) (- y z))
       (if (<= z -7.5e-84)
         t_1
         (if (<= z 5.849437786437544e-5) (+ x (/ (* t (- y z)) a)) t_1))))))

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 t_1 = t + ((x - t) / (z / (y - a)));
	double tmp;
	if (z <= -2.946311588664566e-7) {
		tmp = t_1;
	} else if (z <= -5.2e-62) {
		tmp = (t / (a - z)) * (y - z);
	} else if (z <= -7.5e-84) {
		tmp = t_1;
	} else if (z <= 5.849437786437544e-5) {
		tmp = x + ((t * (y - z)) / a);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t + ((x - t) / (z / (y - a)))
    if (z <= (-2.946311588664566d-7)) then
        tmp = t_1
    else if (z <= (-5.2d-62)) then
        tmp = (t / (a - z)) * (y - z)
    else if (z <= (-7.5d-84)) then
        tmp = t_1
    else if (z <= 5.849437786437544d-5) then
        tmp = x + ((t * (y - z)) / a)
    else
        tmp = t_1
    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 t_1 = t + ((x - t) / (z / (y - a)));
	double tmp;
	if (z <= -2.946311588664566e-7) {
		tmp = t_1;
	} else if (z <= -5.2e-62) {
		tmp = (t / (a - z)) * (y - z);
	} else if (z <= -7.5e-84) {
		tmp = t_1;
	} else if (z <= 5.849437786437544e-5) {
		tmp = x + ((t * (y - z)) / a);
	} else {
		tmp = t_1;
	}
	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):
	t_1 = t + ((x - t) / (z / (y - a)))
	tmp = 0
	if z <= -2.946311588664566e-7:
		tmp = t_1
	elif z <= -5.2e-62:
		tmp = (t / (a - z)) * (y - z)
	elif z <= -7.5e-84:
		tmp = t_1
	elif z <= 5.849437786437544e-5:
		tmp = x + ((t * (y - z)) / a)
	else:
		tmp = t_1
	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)
	t_1 = Float64(t + Float64(Float64(x - t) / Float64(z / Float64(y - a))))
	tmp = 0.0
	if (z <= -2.946311588664566e-7)
		tmp = t_1;
	elseif (z <= -5.2e-62)
		tmp = Float64(Float64(t / Float64(a - z)) * Float64(y - z));
	elseif (z <= -7.5e-84)
		tmp = t_1;
	elseif (z <= 5.849437786437544e-5)
		tmp = Float64(x + Float64(Float64(t * Float64(y - z)) / a));
	else
		tmp = t_1;
	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)
	t_1 = t + ((x - t) / (z / (y - a)));
	tmp = 0.0;
	if (z <= -2.946311588664566e-7)
		tmp = t_1;
	elseif (z <= -5.2e-62)
		tmp = (t / (a - z)) * (y - z);
	elseif (z <= -7.5e-84)
		tmp = t_1;
	elseif (z <= 5.849437786437544e-5)
		tmp = x + ((t * (y - z)) / a);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(t + N[(N[(x - t), $MachinePrecision] / N[(z / N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.946311588664566e-7], t$95$1, If[LessEqual[z, -5.2e-62], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -7.5e-84], t$95$1, If[LessEqual[z, 5.849437786437544e-5], N[(x + N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Target

Original	25.1
Target	12.2
Herbie	20.3

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

Derivation

1. Split input into 3 regimes

2. if z < -2.94631158866456581e-7 or -5.1999999999999999e-62 < z < -7.50000000000000026e-84 or 5.84943778643754427e-5 < z

   1. Initial program 36.8

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

   2. Simplified 20.5

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

   3. Taylor expanded in z around -inf 28.0

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

   4. Simplified 18.5

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

   if -2.94631158866456581e-7 < z < -5.1999999999999999e-62

   1. Initial program 15.7

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

   2. Simplified 9.7

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

   3. Taylor expanded in t around inf 36.6

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

   4. Simplified 37.3

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

   if -7.50000000000000026e-84 < z < 5.84943778643754427e-5

   1. Initial program 8.3

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

   2. Applied egg-rr 7.1

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

   3. Taylor expanded in a around inf 16.5

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

   4. Simplified 16.5

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

   5. Taylor expanded in t around inf 20.9

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

   6. Simplified 20.9

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

4. Final simplification 20.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.946311588664566 \cdot 10^{-7}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{elif}\;z \leq 5.849437786437544 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{t \cdot \left(y - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	33.2
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{t \cdot y}{a}\\ \mathbf{if}\;z \leq -0.011005571285354716:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -9.56251749112683 \cdot 10^{-27}:\\ \;\;\;\;\frac{t}{\frac{a - z}{y}}\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 5.849437786437544 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \mathbf{elif}\;z \leq 1.4677918117502028 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 2
Error	25.3
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{t \cdot \left(y - z\right)}{a}\\ t_2 := t + x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -0.011005571285354716:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.2219353032293872 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.849437786437544 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	32.3
Cost	976

\[\begin{array}{l} t_1 := x + \frac{t \cdot y}{a}\\ \mathbf{if}\;z \leq -0.011005571285354716:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 5.849437786437544 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \mathbf{elif}\;z \leq 1.4677918117502028 \cdot 10^{+180}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 4
Error	26.8
Cost	976

\[\begin{array}{l} t_1 := t + x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1.159473854595549 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.56251749112683 \cdot 10^{-27}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.849437786437544 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.8
Cost	972

\[\begin{array}{l} t_1 := t + x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -2.1076940382233805 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;z \leq 5.849437786437544 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{t \cdot \left(y - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	23.0
Cost	972

\[\begin{array}{l} t_1 := t + \frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -0.011005571285354716:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;z \leq 5.849437786437544 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{t \cdot \left(y - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	37.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5612807138919113 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -5.357022756327088 \cdot 10^{-44}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-171}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;a \leq 5.3381675873684524 \cdot 10^{-33}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	37.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5612807138919113 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -5.357022756327088 \cdot 10^{-44}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-171}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;a \leq 5.3381675873684524 \cdot 10^{-33}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	37.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5612807138919113 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -5.357022756327088 \cdot 10^{-44}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-171}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;a \leq 5.3381675873684524 \cdot 10^{-33}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	30.1
Cost	844

\[\begin{array}{l} t_1 := t - y \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -1.159473854595549 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 7.342607652985869 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	36.4
Cost	716

\[\begin{array}{l} \mathbf{if}\;a \leq -1.126403446567394 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-171}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq -6.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;a \leq 5.3381675873684524 \cdot 10^{-33}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	36.4
Cost	716

\[\begin{array}{l} \mathbf{if}\;a \leq -1.126403446567394 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-171}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;a \leq 5.3381675873684524 \cdot 10^{-33}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	32.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.011005571285354716:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.4677918117502028 \cdot 10^{+180}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 14
Error	30.1
Cost	712

\[\begin{array}{l} t_1 := t - y \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.342607652985869 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	36.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 4.263663020907866 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 16
Error	45.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022228 
(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))))