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

Average Error: 24.3 → 5.2

Time: 9.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ t_3 := \frac{a - t}{z - t}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-223}:\\ \;\;\;\;t_1 + x \cdot \left(1 - {t_3}^{-1}\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1 + x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_3} + x \cdot \left(\frac{t - z}{a - t} + 1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t))))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t))))
        (t_3 (/ (- a t) (- z t))))
   (if (<= t_2 -2e-223)
     (+ t_1 (* x (- 1.0 (pow t_3 -1.0))))
     (if (<= t_2 0.0)
       (+ t_1 (* x (/ (- z a) t)))
       (+ (/ y t_3) (* x (+ (/ (- t z) (- a t)) 1.0)))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double t_3 = (a - t) / (z - t);
	double tmp;
	if (t_2 <= -2e-223) {
		tmp = t_1 + (x * (1.0 - pow(t_3, -1.0)));
	} else if (t_2 <= 0.0) {
		tmp = t_1 + (x * ((z - a) / t));
	} else {
		tmp = (y / t_3) + (x * (((t - z) / (a - t)) + 1.0));
	}
	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 - x) * (z - t)) / (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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = y * ((z - t) / (a - t))
    t_2 = x + (((y - x) * (z - t)) / (a - t))
    t_3 = (a - t) / (z - t)
    if (t_2 <= (-2d-223)) then
        tmp = t_1 + (x * (1.0d0 - (t_3 ** (-1.0d0))))
    else if (t_2 <= 0.0d0) then
        tmp = t_1 + (x * ((z - a) / t))
    else
        tmp = (y / t_3) + (x * (((t - z) / (a - t)) + 1.0d0))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double t_3 = (a - t) / (z - t);
	double tmp;
	if (t_2 <= -2e-223) {
		tmp = t_1 + (x * (1.0 - Math.pow(t_3, -1.0)));
	} else if (t_2 <= 0.0) {
		tmp = t_1 + (x * ((z - a) / t));
	} else {
		tmp = (y / t_3) + (x * (((t - z) / (a - t)) + 1.0));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (a - t))
	t_2 = x + (((y - x) * (z - t)) / (a - t))
	t_3 = (a - t) / (z - t)
	tmp = 0
	if t_2 <= -2e-223:
		tmp = t_1 + (x * (1.0 - math.pow(t_3, -1.0)))
	elif t_2 <= 0.0:
		tmp = t_1 + (x * ((z - a) / t))
	else:
		tmp = (y / t_3) + (x * (((t - z) / (a - t)) + 1.0))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	t_3 = Float64(Float64(a - t) / Float64(z - t))
	tmp = 0.0
	if (t_2 <= -2e-223)
		tmp = Float64(t_1 + Float64(x * Float64(1.0 - (t_3 ^ -1.0))));
	elseif (t_2 <= 0.0)
		tmp = Float64(t_1 + Float64(x * Float64(Float64(z - a) / t)));
	else
		tmp = Float64(Float64(y / t_3) + Float64(x * Float64(Float64(Float64(t - z) / Float64(a - t)) + 1.0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (a - t));
	t_2 = x + (((y - x) * (z - t)) / (a - t));
	t_3 = (a - t) / (z - t);
	tmp = 0.0;
	if (t_2 <= -2e-223)
		tmp = t_1 + (x * (1.0 - (t_3 ^ -1.0)));
	elseif (t_2 <= 0.0)
		tmp = t_1 + (x * ((z - a) / t));
	else
		tmp = (y / t_3) + (x * (((t - z) / (a - t)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-223], N[(t$95$1 + N[(x * N[(1.0 - N[Power[t$95$3, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$1 + N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / t$95$3), $MachinePrecision] + N[(x * N[(N[(N[(t - z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	24.3
Target	9.3
Herbie	5.2

\[\begin{array}{l} \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1.9999999999999999e-223

   1. Initial program 21.3

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

   2. Simplified 9.8

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

   3. Taylor expanded in x around -inf 14.6

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

   4. Simplified 4.7

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

   5. Applied egg-rr 4.7

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

   if -1.9999999999999999e-223 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 53.4

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

   2. Simplified 54.1

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

   3. Taylor expanded in x around -inf 15.7

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

   4. Simplified 29.7

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

   5. Taylor expanded in t around inf 6.8

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

   6. Simplified 7.0

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

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

   1. Initial program 21.1

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

   2. Simplified 10.7

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

   3. Taylor expanded in x around -inf 14.9

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

   4. Simplified 5.2

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

   5. Applied egg-rr 5.2

      \[\leadsto \color{blue}{\frac{y}{\frac{a - t}{z - t}}} - x \cdot \left(\frac{z - t}{a - t} + -1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2 \cdot 10^{-223}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t} + x \cdot \left(1 - {\left(\frac{a - t}{z - t}\right)}^{-1}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;y \cdot \frac{z - t}{a - t} + x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}} + x \cdot \left(\frac{t - z}{a - t} + 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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