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

Average Accuracy: 61.5% → 91.6%

Time: 26.3s

Precision: binary64

Cost: 3145

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (or (<= t_1 -1e-260) (not (<= t_1 0.0)))
     (+ (* (- t x) (/ y (- a z))) (- x (* (- t x) (/ z (- a z)))))
     (+ (- t (* (/ a z) (- x t))) (* (/ y z) (- x t))))))

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 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if ((t_1 <= -1e-260) || !(t_1 <= 0.0)) {
		tmp = ((t - x) * (y / (a - z))) + (x - ((t - x) * (z / (a - z))));
	} else {
		tmp = (t - ((a / z) * (x - t))) + ((y / z) * (x - t));
	}
	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 = x + (((y - z) * (t - x)) / (a - z))
    if ((t_1 <= (-1d-260)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = ((t - x) * (y / (a - z))) + (x - ((t - x) * (z / (a - z))))
    else
        tmp = (t - ((a / z) * (x - t))) + ((y / z) * (x - t))
    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 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if ((t_1 <= -1e-260) || !(t_1 <= 0.0)) {
		tmp = ((t - x) * (y / (a - z))) + (x - ((t - x) * (z / (a - z))));
	} else {
		tmp = (t - ((a / z) * (x - t))) + ((y / z) * (x - t));
	}
	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 = x + (((y - z) * (t - x)) / (a - z))
	tmp = 0
	if (t_1 <= -1e-260) or not (t_1 <= 0.0):
		tmp = ((t - x) * (y / (a - z))) + (x - ((t - x) * (z / (a - z))))
	else:
		tmp = (t - ((a / z) * (x - t))) + ((y / z) * (x - t))
	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(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if ((t_1 <= -1e-260) || !(t_1 <= 0.0))
		tmp = Float64(Float64(Float64(t - x) * Float64(y / Float64(a - z))) + Float64(x - Float64(Float64(t - x) * Float64(z / Float64(a - z)))));
	else
		tmp = Float64(Float64(t - Float64(Float64(a / z) * Float64(x - t))) + Float64(Float64(y / z) * Float64(x - t)));
	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 = x + (((y - z) * (t - x)) / (a - z));
	tmp = 0.0;
	if ((t_1 <= -1e-260) || ~((t_1 <= 0.0)))
		tmp = ((t - x) * (y / (a - z))) + (x - ((t - x) * (z / (a - z))));
	else
		tmp = (t - ((a / z) * (x - t))) + ((y / z) * (x - t));
	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[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-260], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(t - x), $MachinePrecision] * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(t - x), $MachinePrecision] * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - N[(N[(a / z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	61.5%
Target	81.5%
Herbie	91.6%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -9.99999999999999961e-261 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 66.4%

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

   2. Applied egg-rr 91.3%

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

      [Start] 66.4	\[ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
      +-commutative [=>] 66.4	\[ \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]
      associate-/l* [=>] 83.7	\[ \color{blue}{\frac{y - z}{\frac{a - z}{t - x}}} + x \]
      div-sub [=>] 83.7	\[ \color{blue}{\left(\frac{y}{\frac{a - z}{t - x}} - \frac{z}{\frac{a - z}{t - x}}\right)} + x \]
      associate-+l- [=>] 86.2	\[ \color{blue}{\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)} \]
      associate-/r/ [=>] 86.3	\[ \color{blue}{\frac{y}{a - z} \cdot \left(t - x\right)} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right) \]
      associate-/r/ [=>] 91.3	\[ \frac{y}{a - z} \cdot \left(t - x\right) - \left(\color{blue}{\frac{z}{a - z} \cdot \left(t - x\right)} - x\right) \]

   if -9.99999999999999961e-261 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 10.6%

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

   2. Simplified 9.9%

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

      [Start] 10.6	\[ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
      +-commutative [=>] 10.6	\[ \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]
      associate-*r/ [<=] 9.5	\[ \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]
      *-commutative [<=] 9.5	\[ \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]
      fma-def [=>] 9.9	\[ \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   3. Taylor expanded in z around inf 94.2%

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

   4. Simplified 94.4%

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

      [Start] 94.2	\[ -1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \left(t + \frac{a \cdot \left(t - x\right)}{z}\right) \]
      +-commutative [=>] 94.2	\[ \color{blue}{\left(t + \frac{a \cdot \left(t - x\right)}{z}\right) + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}} \]
      mul-1-neg [=>] 94.2	\[ \left(t + \frac{a \cdot \left(t - x\right)}{z}\right) + \color{blue}{\left(-\frac{y \cdot \left(t - x\right)}{z}\right)} \]
      unsub-neg [=>] 94.2	\[ \color{blue}{\left(t + \frac{a \cdot \left(t - x\right)}{z}\right) - \frac{y \cdot \left(t - x\right)}{z}} \]
      associate-/l* [=>] 88.4	\[ \left(t + \color{blue}{\frac{a}{\frac{z}{t - x}}}\right) - \frac{y \cdot \left(t - x\right)}{z} \]
      associate-/r/ [=>] 94.2	\[ \left(t + \color{blue}{\frac{a}{z} \cdot \left(t - x\right)}\right) - \frac{y \cdot \left(t - x\right)}{z} \]
      associate-/l* [=>] 88.6	\[ \left(t + \frac{a}{z} \cdot \left(t - x\right)\right) - \color{blue}{\frac{y}{\frac{z}{t - x}}} \]
      associate-/r/ [=>] 94.4	\[ \left(t + \frac{a}{z} \cdot \left(t - x\right)\right) - \color{blue}{\frac{y}{z} \cdot \left(t - x\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

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

Alternatives

Alternative 1
Accuracy	89.1%
Cost	2889

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

Alternative 2
Accuracy	89.1%
Cost	2633

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

Alternative 3
Accuracy	74.1%
Cost	1232

\[\begin{array}{l} t_1 := t - \frac{y}{\frac{z}{t - x}}\\ t_2 := x + \frac{y - z}{\frac{a - z}{t}}\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-226}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	77.6%
Cost	1232

\[\begin{array}{l} t_1 := x + \frac{y - z}{\frac{a - z}{t}}\\ t_2 := t + \frac{y - a}{z} \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-193}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+172}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	47.5%
Cost	1040

\[\begin{array}{l} t_1 := \frac{-t}{\frac{a - z}{z}}\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{+146}:\\ \;\;\;\;x - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq -5.7 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-271}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 6
Accuracy	47.9%
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-185}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-270}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-23}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	47.7%
Cost	976

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-14}:\\ \;\;\;\;x - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq -7.8 \cdot 10^{-185}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-35}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 8
Accuracy	64.9%
Cost	972

\[\begin{array}{l} t_1 := x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{-184}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-62}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	67.1%
Cost	972

\[\begin{array}{l} t_1 := x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -9.5 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.4 \cdot 10^{-63}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+19}:\\ \;\;\;\;t - \frac{y}{\frac{z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	42.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+146}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-184}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.35 \cdot 10^{-270}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	63.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{+149} \lor \neg \left(a \leq 1.3 \cdot 10^{+19}\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array} \]

Alternative 12
Accuracy	59.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+146}:\\ \;\;\;\;x - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+141}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 13
Accuracy	43.0%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-14}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-172}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 14
Accuracy	43.0%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-14}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-172}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-251}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 15
Accuracy	43.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+146}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Accuracy	28.6%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023138 
(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))))