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

Average Error: 24.1 → 6.9

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t_2 \leq -6.127736774225424 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(t + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ x (* (- t x) (/ (- y z) (- a z)))))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 -6.127736774225424e-272)
     t_1
     (if (<= t_2 0.0)
       (-
        (+ (/ (* x y) z) (+ t (/ (* t a) z)))
        (+ (/ (* y t) z) (/ (* x a) z)))
       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 = x + ((t - x) * ((y - z) / (a - z)));
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -6.127736774225424e-272) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = (((x * y) / z) + (t + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = x + ((t - x) * ((y - z) / (a - z)))
    t_2 = x + (((y - z) * (t - x)) / (a - z))
    if (t_2 <= (-6.127736774225424d-272)) then
        tmp = t_1
    else if (t_2 <= 0.0d0) then
        tmp = (((x * y) / z) + (t + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z))
    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 = x + ((t - x) * ((y - z) / (a - z)));
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -6.127736774225424e-272) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = (((x * y) / z) + (t + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z));
	} 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 = x + ((t - x) * ((y - z) / (a - z)))
	t_2 = x + (((y - z) * (t - x)) / (a - z))
	tmp = 0
	if t_2 <= -6.127736774225424e-272:
		tmp = t_1
	elif t_2 <= 0.0:
		tmp = (((x * y) / z) + (t + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z))
	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(x + Float64(Float64(t - x) * Float64(Float64(y - z) / Float64(a - z))))
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= -6.127736774225424e-272)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * y) / z) + Float64(t + Float64(Float64(t * a) / z))) - Float64(Float64(Float64(y * t) / z) + Float64(Float64(x * a) / z)));
	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 = x + ((t - x) * ((y - z) / (a - z)));
	t_2 = x + (((y - z) * (t - x)) / (a - z));
	tmp = 0.0;
	if (t_2 <= -6.127736774225424e-272)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = (((x * y) / z) + (t + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z));
	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[(x + N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -6.127736774225424e-272], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] + N[(t + N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision] + N[(N[(x * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.1
Target	11.9
Herbie	6.9

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -6.127736774225424e-272 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 21.0

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

   2. Simplified 10.3

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

   3. Applied clear-num_binary64 10.6

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

   4. Applied fma-udef_binary64 10.6

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

   5. Simplified 10.3

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

   6. Applied associate-/r/_binary64 7.2

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

   if -6.127736774225424e-272 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 58.4

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

   2. Simplified 58.3

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

   3. Taylor expanded in z around inf 3.2

      \[\leadsto \color{blue}{\left(\frac{y \cdot x}{z} + \left(t + \frac{a \cdot t}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{a \cdot x}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -6.127736774225424 \cdot 10^{-272}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(t + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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))))